The diagram shows parts of the graphs of \(y=2+5\mathrm{e}^x\) and \(y=4-3\mathrm{e}^{2x}\).

Find the area of the shaded region. Give your answer in the form \(a+b\ln 3\), where \(a\) and \(b\) are exact constants.

The diagram shows part of each of the curves \(y=12-x^2\) and \(y=x^4-4x^2+8\).

Find the area of the shaded region enclosed by the two curves.

The curve \(y=x^2-9x+18\) has a normal at the point where \(x=5\). This normal meets the curve again. Find the area of the shaded region enclosed by the normal and the curve.

The point \(A\) with \(x\)-coordinate \(2\) lies on the curve \(y=\sqrt{4x+1}\). The diagram shows part of this curve and the tangent to the curve at \(A\).

Find the area of the shaded region enclosed by the curve, the tangent and the \(x\)-axis.

д№™

The diagram shows part of the curve \(y=\frac{3}{x+1}-\frac{x-2}{x}\). The points \(A\) and \(B\) lie on the curve such that the \(x\)-coordinate of \(A\) is 1 and the \(x\)-coordinate of \(B\) is 2 . (a) Find the \(y\)-coordinates of \(A\) and \(B\).

(b) Show that the area of the shaded region enclosed by the line \(A B\) and the curve is \(\frac{a}{4}-\ln \frac{b}{2}\), where \(a\) and \(b\) are integers to be found.

The diagram shows part of the curve \(y=\frac{15}{x}-\frac{5}{x^{2}}\). The curve meets the \(x\)-axis at the point \(A\). The curve has a maximum at the point \(B\).

Find the area of the shaded region enclosed by the line \(A B\) and the curve. Give your answer in exact form.

The diagram shows part of the curve \(y=\frac{5}{x+1}+2\) and part of the line \(y=2x+1\) intersecting at the point \(A\).

(a) Find the coordinates of \(A\).

(b) Find the exact area of the shaded region.

The diagram shows part of the curve \(y=4+2 \sin 3 x\) and the straight line \(A B\). The points \(A\) and \(B\) lie on the curve. The \(x\)-coordinate of \(A\) is \(\frac{\pi}{18}\) and the \(x\)-coordinate of \(B\) is \(\frac{\pi}{3}\). Find the area of the shaded region, giving your answer in exact form.

Continuation of working space for Question 11.

The diagram shows part of the curve \(y=\frac{4}{2 x+1}\) and the straight line \(2 y=6 x+1\). Find the area of the shaded region, giving your answer in the form \(\ln a+b\), where \(a\) is an integer and \(b\) is a rational number.

Continuation of working space for question 9.

(a) Solve the equation \(\sin 4 x=\frac{1}{2}\) for \(0 \leqslant x \leqslant \frac{\pi}{4}\), giving your answers in terms of \(\pi\).

(b)

The diagram shows parts of the graphs of \(y=\sin 4 x\) and \(y=\frac{1}{2}\). Find the exact area of the shaded region enclosed by the curve and the line.

The diagram shows part of the curve \(y=32 x-4 x^{2}-48\) and the line \(A B\). The curve and the line \(A B\) meet the \(x\)-axis at \(A\) and meet again at the point \(B(5,12)\). The line \(C D\) extended is parallel to the \(y\)-axis and passes through the maximum point of the curve. Find the area of the shaded region.

Continuation of working space for Question 9.

The diagram shows a sketch of part of the curve \(y=4+(3 x-1)^{-1}\) and the line \(x=9\). The point \(A\) has \(x\)-coordinate 1 . The tangent to the curve at \(A\) meets the \(x\)-axis at the point \(B\). Find the area of the shaded region.

The diagram shows part of the curve

\(y=2-\frac{3}{x-1}\)

and the straight line

\(6y=9-2x.\)

The curve intersects the \(x\)-axis at point \(A\), and the line at point \(B\). The line intersects the \(x\)-axis at point \(C\). Find the area of the shaded region \(ABC\), giving your answer in the form

\(p+\ln q,\)

where \(p\) and \(q\) are rational numbers.

The diagram shows part of the curve

\(y=3+\frac{4}{2x+1}\)

and the straight line

\(3y=2x+6.\)

Find the area of the shaded region, giving your answer in exact form.

(a) The diagram shows the curve \(y=6x-x^2\), for \(0\leq x\leq5\), and the line \(y=x\). Find the area of the shaded region.

(b) Find:

(i)

\(\int\left(\frac{1}{(2x-6)^3}+\cos x\right)\,dx.\)

(ii)

\(\int\frac{(x^4+1)^2}{2x}\,dx.\)

(a) Show that

\(\int_1^8\frac{x+4}{\sqrt x}\,dx=36.6.\)

(b) The diagram shows the line

\(10y=7-3x\)

and the curve

\(y=\frac{1}{3x+4}.\)

The line and curve intersect at the point \(A\).

Verify that the \(y\)-coordinate of \(A\) is \(0.1\), and calculate the shaded area.

The diagram shows part of the graphs of

\(y=6+e^{4x-5} \quad\text{and}\quad x=2.\)

The line \(x=2\) meets the curve at the point \(B(2,b)\), and the line \(AB\) is parallel to the \(x\)-axis. Find the area of the shaded region.

The diagram shows part of the curves \(y=\mathrm{e}^{x/2}\) and \(y=\cos5x\), and part of the line \(x=\frac{\pi}{4}\). The curves intersect at \(A\). The curve \(y=\cos5x\) cuts the \(x\)-axis at \(B\). The line \(x=\frac{\pi}{4}\) cuts the \(x\)-axis at \(C\) and the curve \(y=\mathrm{e}^{x/2}\) at \(D\). Find the exact area of the shaded region \(ABCD\).

The diagram shows part of the line \(y=1\) and one complete period of the curve \(y=1+\cos x\), where \(x\) is in radians. The line \(PQ\) is a tangent to the curve at \(P\) and at \(Q\). The line \(QR\) is parallel to the \(y\)-axis. Area \(A\) is enclosed by the line \(y=1\) and the curve. Area \(B\) is enclosed by the line \(y=1\), the line \(PQ\) and the curve. Given that area \(A\) : area \(B\) is \(1:k\), find the exact value of \(k\).

The diagram shows part of the curve \(y=3+2x-x^2\). The point \(A\) on the curve has \(x\)-coordinate \(1.5\). The tangent to the curve at \(A\) meets the \(x\)-axis at \(B\). The curve meets the \(x\)-axis at \(C\). Find the area of the shaded region.

The polynomial \(\mathrm p(x)=2x^3-3x^2-x+1\) has a factor \(2x-1\).

(a) Find \(\mathrm p(x)\) in the form \((2x-1)\mathrm q(x)\), where \(\mathrm q(x)\) is a quadratic factor.

The diagram shows the graph of \(y=\dfrac1x\), for \(x\gt 0\), and the graph of \(y=-2x^2+3x+1\). The curves intersect at the points \(A\) and \(B\).

(b) Using your answer to part (a), find the exact \(x\)-coordinate of \(A\) and of \(B\).

(c) Find the exact area of the shaded region.

The diagram shows part of the curve

\(y=(9-x)(x-3)\)

and the line \(y=k-3\), where \(k\gt 3\). The line through the maximum point of the curve, parallel to the \(y\)-axis, meets the \(x\)-axis at \(A\). The curve meets the \(x\)-axis at \(B\), and the line \(y=k-3\) meets the curve at the point \(C(k,k-3)\). Find the area of the shaded region.

Do not use a calculator in this question.

The diagram shows part of the curve

\(y=\frac1{2x+1}\)

and part of the line

\(5y=x-1.\)

The curve meets the \(y\)-axis at \(A\). The line meets the \(x\)-axis at \(B\). The line and curve intersect at point \(C\).

(a)

(i) Find the coordinates of \(A\) and \(B\).

(ii) Verify that the \(x\)-coordinate of \(C\) is \(2\).

(b) Find the exact area of the shaded region.

(a) Solve the inequality

\(2x^2-17x+21\leq0.\)

(b) Hence find the area enclosed between the curve

\(y=2x^2-17x+21\)

and the \(x\)-axis.

The diagram shows the graph of the curve

\(y=\frac{1}{(x+2)^2}+\frac{3}{x+2}\)

for \(x\gt -2\). The points \(A\) and \(B\) lie on the curve such that the \(x\)-coordinates of \(A\) and \(B\) are \(-1\) and \(2\) respectively.

(a) Find the exact \(y\)-coordinates of \(A\) and of \(B\).

(b) Find the area of the shaded region enclosed by the line \(AB\) and the curve, giving your answer in the form \(\frac pq-\ln r\), where \(p\), \(q\) and \(r\) are integers.

The diagram shows part of the curve

\(y=\frac5{x-1}+2x,\)

and the straight lines \(x=4\) and \(2y=9x\).

(a) Find the coordinates of the stationary point on the curve

\(y=\frac5{x-1}+2x.\)

(b) Given that the curve and the line \(2y=9x\) intersect at the point \((2,9)\), find the area of the shaded region.

The diagram shows the line \(2x+y=-5\) and the curve \(xy+3=0\). The line intersects the \(x\)-axis at \(A\) and the curve at \(B\). The point \(C\) lies on the curve, and \(D\) has coordinates \((1,0)\). The line \(CD\) is parallel to the \(y\)-axis.

(a) Find the coordinates of \(A\) and \(B\).

(b) Find the area of the shaded region, giving your answer in the form \(p+\ln q\), where \(p\) and \(q\) are positive integers.

The diagram shows part of the curve \(xy=2\) intersecting the straight line \(y=5x-3\) at the point \(A\). The straight line meets the \(x\)-axis at the point \(B\). The point \(C\) lies on the \(x\)-axis and the point \(D\) lies on the curve such that the line \(CD\) has equation \(x=3\).

Find the exact area of the shaded region, giving your answer in the form \(p+\ln q\), where \(p\) and \(q\) are constants.

The diagram shows part of the graphs of

\(y=4x^{\frac23} \qquad\text{and}\qquad y=(x-3)^2.\)

The graph of \(y=(x-3)^2\) meets the \(x\)-axis at the point \(A(a,0)\), and the two graphs intersect at the point \(B(b,4)\).

(a) Find the value of \(a\) and of \(b\).

(b) Find the area of the shaded region.

(a) Show that

\(\frac{1}{x+1}+\frac{2}{3x+10}\)

can be written as

\(\frac{5x+12}{3x^2+13x+10}.\)

(b) The diagram shows part of the curve

\(y=\frac{5x+12}{3x^2+13x+10},\)

the line \(x=2\) and a straight line of gradient \(1\). The curve intersects the \(y\)-axis at the point \(P\). The line of gradient \(1\) passes through \(P\) and intersects the \(x\)-axis at the point \(Q\). Find the area of the shaded region, giving your answer in the form

\(a+\frac23\ln(b\sqrt3),\)

where \(a\) and \(b\) are constants.

The diagram shows the curve \(y=1+x+5\sqrt{x}\) and the straight line \(y-3x=3\). The curve and line intersect at the points \(A\) and \(B\). The lines \(BC\) and \(AD\) are perpendicular to the \(x\)-axis.

(i) Using the substitution \(u^2=x\), or otherwise, find the coordinates of \(A\) and of \(B\). You must show all your working.

(ii) Find the area of the shaded region, showing all your working.

The diagram shows the curve \(y=3x^2-2x+1\) and the straight line \(y=2x+5\) intersecting at the points \(P\) and \(Q\). Find the area of the shaded region.

The diagram shows part of the graph of \(y=2+\cos3x\) and the straight line \(y=1.5\). Find the exact area of the shaded region bounded by the curve and the straight line. You must show all your working.

The diagram shows part of the curve \(y=(x+2)^2(1-3x)\). The curve has a minimum point \(A\) and a maximum point \(B\). The curve intersects the \(y\)-axis at \(C\) and the \(x\)-axis at \(D\).

(i) Find the \(x\)-coordinates of \(A\) and \(B\).

(ii) Find the coordinates of \(C\) and \(D\).

(iii) Find the area of the shaded region.

The diagram shows part of the graph of

\(y=16x+\frac{27}{x^2},\)

which has a minimum at \(A\).

(i) Find the coordinates of \(A\).

The points \(P\) and \(Q\) lie on the curve \(y=16x+\dfrac{27}{x^2}\) and have \(x\)-coordinates \(1\) and \(3\) respectively.

(ii) Find the area enclosed by the curve and the line \(PQ\). You must show all your working.

The diagram shows the graph of the curve

\(y=\frac{e^{4x}+3}{8}.\)

The curve meets the \(y\)-axis at the point \(A\). The normal to the curve at \(A\) meets the \(x\)-axis at the point \(B\). Find the area of the shaded region enclosed by the curve, the line \(AB\) and the line through \(B\) parallel to the \(y\)-axis. Give your answer in the form \(\dfrac ea\), where \(a\) is a constant.

The diagram shows part of the graph of

\(y=16x+\frac{27}{x^2},\)

which has a minimum at \(A\).

(i) Find the coordinates of \(A\).

The points \(P\) and \(Q\) lie on the curve \(y=16x+\dfrac{27}{x^2}\) and have \(x\)-coordinates \(1\) and \(3\) respectively.

(ii) Find the area enclosed by the curve and the line \(PQ\). You must show all your working.

The diagram shows the curve \(y=12+x-x^2\) intersecting the line \(y=x+8\) at the points \(A\) and \(B\).

(i) Find the coordinates of the points \(A\) and \(B\).

(ii) Find \(\displaystyle\int(12+x-x^2)\,dx\).

(iii) Showing all your working, find the area of the shaded region.

The diagram shows part of the curve

\(y=x+e^{5-2x},\)

the normal to the curve at the point \(A\), and the line \(x=5\). The normal to the curve at \(A\) meets the \(y\)-axis at the point \(B\). The \(x\)-coordinate of \(A\) is \(2.5\).

(i) Find the equation of the normal \(AB\).

(ii) Showing all your working, find the area of the shaded region.

The diagram shows part of the curve \(y=2\sqrt{x}\). The normal to the curve at the point \(A(4,4)\) meets the \(x\)-axis at the point \(B\).

(i) Find the equation of the line \(AB\).

(ii) Find the coordinates of \(B\).

(iii) Showing all your working, find the area of the shaded region.

The diagram shows part of the curve \(y=x^3+4x^2-5x+5\) and the line \(y=5\).

The curve and the line intersect at the points \(A\), \(B\) and \(C\).

The points \(D\) and \(E\) are on the \(x\)-axis and the lines \(AE\) and \(CD\) are parallel to the \(y\)-axis.

(i) Find \(\int(x^3+4x^2-5x+5)\,dx\).

(ii) Find the area of each of the rectangles \(OEAB\) and \(OBCD\).

(iii) Hence calculate the total area of the shaded regions enclosed between the line and the curve. You must show all your working.

The diagram shows part of the curve \(y=5+\sqrt{10x}\) and the line \(4y=5x+20\). The line and curve intersect at the points \(P(0,5)\) and \(Q\). The line \(QR\) is parallel to the \(y\)-axis.

(i) Find the coordinates of \(Q\).

(ii) Find the area of the shaded region. You must show all your working.

The diagram shows the curve \(y=4+3x-x^2\) intersecting the positive \(x\)-axis at the point \(A\). The line \(y=mx+8\) is a tangent to the curve at the point \(B\). Find

(i) the coordinates of \(A\),

(ii) the value of \(m\),

(iii) the coordinates of \(B\),

(iv) the area of the shaded region, showing all your working.