Further Integration (DP IB Applications & Interpretation (AI): HL): Exam Questions

Consider the function  $f$ defined by $f\left(x\right)=\left(x^2-x-2\right)\left(x-5\right),-2\le x\le 4.$

Find the coordinates of the points where the graph of  $y=f\left(x\right)$ intercepts the $x$-axis.

Find the indefinite integral

$\int \left(x^2-x-2\right)\left(x-5\right)dx$

Use your answer to part (b) to calculate the area of the region enclosed by the graph of  $y=f\left(x\right)$ and the $x$-axis.

Find the indefinite integral for

$\int \cos \left(\frac{x}{2}\right)dx$

Find the indefinite integral for

$\int 5e^{3x}dx$

Find an expression for $y$ given that

$\frac{dy}{dx}=\sin \left(x-\frac{\mathrm{\pi }}{3}\right)$

Find the indefinite integral

$\int \frac{3}{2x}dx$

Find an expression for given that

 $\frac{dy}{dx}=e^{2x+3}+2$

 and also that $y=5$ when $x=-\frac{3}{2}$ .

Find the indefinite integral for

$\int \left(\frac{1}{4}\sqrt{x}-\frac{2}{\sqrt[3]{x}}\right)dx$

Find the indefinite integral for

$\int \frac{x\frac{5}{6}-5x\frac{7}{3}}{x\sqrt{x}}dx$

Consider the function $f\left(x\right)=\ln \left(3x^2-12x+1\right)$.

(i) Find  $f\text{'}\left(x\right)$.

(ii) Hence, find 

$\int \frac{16-8x}{3x^2-12x+1}dx$

Let $g\text{'}\left(x\right)=\left(x^2-5x+6\right)\sin \left(2x^3-15x^2+36x-\frac{\mathrm{\pi }}{3}\right)$

Find $g\left(x\right)$ given that  $g\left(0\right)=1$.

Work out the following indefinite integrals:

$\int \frac{2}{3{\cos }^{2}3x}dx$

$\int \frac{7\tan \left(\frac{x}{2}\right)}{6\sin \frac{x}{2}\cos \frac{x}{2}}dx$

Use definite integration to find the exact value of

 ${\int }_2^5\frac{x+1}{x^2+2x-5}dx$ 

giving your final answer in as simple a form as possible.

The diagram below shows the graph of the function $f$ which is defined by

$f\left(x\right)=-x\cos x,0\le x\le \frac{3\mathrm{\pi }}{2}$

The shaded region in the diagram is the region enclosed by the $x$-axis and the graph of $y=f\left(x\right)$. 

Explain why the area of the shaded region is not equal to

${\int }_0^{\frac{3\mathrm{\pi }}{2}}-x\cos xdx$

Find the area of the entire shaded region.

Find the individual areas of the parts of the shaded region (i) above and (ii) below the x-axis.

The shaded region in the diagram below depicts the design for a new company logo.  The upper border of the logo is formed by a part of the curve with equation $y=\frac{9{\mathrm{\pi }}^2-16x^2}{32}$,  while the lower border of the logo is formed by a part of the curve with equation  $y=-1+2{\sin }^{2}x$.  The points where the two curves intersect lie on the $x$-axis, as shown.

Find the exact coordinates of 

(i) the points of intersection of the two curves 

(ii) the other points where the lower border of the logo intersects the $x$-axis.

Hence find the area of the company logo.

The following diagram shows a part of the graph of the curve  $y=kx-x^2$,  where $k>0$  is a constant.  The point marked A is the vertex of the curve.  Region $R$ is the region enclosed by the curve and the $x$-axis.  Region $S$ is the region enclosed by the curve, the positive $y$-axis, and the line through point A with gradient zero.

Show that the part of the curve bordering the region  can also be represented by the curve with equation  $x=\frac{1}{2}\left(k-\sqrt{k^2-4y}\right)$.

When region $R$ is rotated $2\mathrm{\pi }$ radians about the $x$-axis, the resultant solid of revolution has a volume equal to $\frac{1296\mathrm{\pi }}{5}{\mathrm{units}}^3$ . 

Find the value of $k$.

Use the result from part (a) to find the area of region $S$.

Use your answer to part (c) to write down the area of region R.

The diagram below shows the cross-section of a miniature goldfish bowl produced by Some Things Fishy, a specialist company supplying products for miniature goldfish enthusiasts.

The glass part of the bowl sits on a solid base, indicated by the shaded region on the diagram.  The cross-section of the glass part of the bowl is symmetrical about the $y$-axis, and may be described by the curve with equation

 $\frac{x^2}{64}+\frac{{\left(y-5\right)}^2}{16}=1$

The dashed horizontal line represents the diameter of the open top of the fishbowl.  All coordinates are expressed in centimetres, and for purposes of answering this question the thickness of the glass sides of the bowl may be regarded as negligible. 

Given that the diameter of the open top of the fishbowl is 15 cm, and that this is less than the diameter of the fishbowl at its widest point, find the capacity of the glass part of the fishbowl.

Consider the function $f$ defined by $f\left(x\right)=\left(x^2-3x+2\right)\left(x+2\right),-3\le x\le \frac{5}{4}.$

Find the indefinite integral

$\int \left(x^2-3x+2\right)\left(x+2\right)dx$

Use your answer to part (a) to calculate the area of the region enclosed by the graph of  $y=f\left(x\right)$ and the $x$-axis.

Find the indefinite integral for

$\int \sin \left(\frac{\sqrt{3}}{2}x\right)dx$

Find the indefinite integral for

$\int \frac{7}{e^{4x-9}}dx$

Find an expression for $y$ given that

$\frac{dy}{dx}=\cos \left(2\left(\frac{\mathrm{\pi }}{8}-x\right)\right)$

Find the indefinite integral

$\int -\frac{7}{5x}dx$

Find an expression for given that

$\frac{dy}{dx}=x{e}^{x^2-2}$

and also that $y=3$ when $x=-\sqrt{2}$

Find the indefinite integral for

$\int \left(\sqrt{3x}+\frac{5}{\sqrt[3]{8x}}\right)dx$

Find the indefinite integral for

$\int \frac{{\left(8x\right)}^{\frac{2}{3}}-5x^{\frac{1}{6}}}{x\sqrt[3]{x}}dx$

Find the indefinite integral

$\int \frac{x+2x^3}{\left(1-x^2\right)\left(2+x^2\right)}dx$

Let  $g\text{'}\left(x\right)=\frac{\cos \left(\ln x\right)}{x}$ for $x>0.$ 

Find  $g\left(x\right)$ given that  $g\left(1\right)=\mathrm{\pi }$.

Show that

$\int \tan xdx=\ln \left|\frac{1}{\cos x}\right|+c$

A curve with equation $y=f\left(x\right)$ is such that

 $\frac{dy}{dx}=\frac{k}{\left(1+\sin \pi x\right)\left(1-\sin \pi x\right)}$ 

where $k$ is a real constant. 

Given that the curve passes through the points $\left(0,-3\right)$ and $\left(-\frac{1}{4},-\mathrm{\pi }\right)$, find $f\left(x\right)$.

Explain why

$\frac{1}{\tan \theta }=\frac{\cos \theta }{\sin \theta }$

Use definite integration, along with the result from part (a), to show that

${\int }_0^{\sqrt{\frac{\mathrm{\pi }}{6}}}\frac{x}{\tan \left(x^2-\frac{2\mathrm{\pi }}{3}\right)}dx=-\frac{1}{2}\ln \left(\frac{\sqrt{3}}{2}\right)$

Using your knowledge of the natural logarithm function, explain (without using your GDC) why the value of the integral found in part (b) is a positive number.

The diagram below shows the graph of the function $f$ which is defined by

$f\left(x\right)=x\sin 2x,0\le x\le \frac{3\mathrm{\pi }}{2}$

The shaded region in the diagram is the region enclosed by the $x$-axis and the graph of $y=f\left(x\right)$.  The three sub-parts of the shaded region are denoted by R, S and T, as shown.

Find the value of

${\int }_0^{\frac{3\mathrm{\pi }}{2}}x\sin 2xdx$

Find the individual areas of each of the three sub-parts R, S and T of the shaded region.

Compare the sum of the answers in part (b) to the answer in part (a) and comment on the result.

The diagram below depicts the design for a new company logo.  The upper border of the logo is formed by a part of the curve with equation  $y=9-x^2$,  while the lower border of the logo is formed by a part of the curve with equation $y=\frac{1}{8}{x}^2-5$.  As shown in the diagram, the logo is divided into a shaded part and an unshaded part by a part of the curve with equation $y=\frac{\left(9-x^2\right)\left(2x-1\right)}{10}$.

Find the percentage of the total area of the logo that is shaded.

The following diagram shows a part of the graph of the curve with equation $y^2=k\left(1600-25{\left(x-8\right)}^2\right)$,  where $k>0$ is a constant.  The point marked A is the vertex of the curve.  Region $R$ is the region enclosed by the curve and the $x$-axis, for the part of the curve where $y$ is non-negative.  Region $S$ is the region enclosed by the curve, the positive $y$-axis, and the line through point A with gradient zero.

When region $R$ is rotated $2\mathrm{\pi }$ radians about the $x$-axis, the resultant solid of revolution has a volume equal to $\frac{12800\mathrm{\pi }}{3}{\mathrm{units}}^3$. 

Find the area of region $S$ by calculating an area between the curve and the $y$-axis.

Find the area of region S by calculating an area between the curve and the $x$-axis.  Confirm that this matches your answer to part (a).

The diagram below shows the cross-section of a goldfish bowl to be produced by Pieseize Manufacturing, a specialist company supplying products for goldfish enthusiasts.

The glass part of the bowl sits on a solid base, indicated by the shaded region on the diagram. The cross-section of the glass part of the bowl is symmetrical about the $y$-axis, and may be described by the curve with equation

$\frac{x^2}{400}+\frac{{\left(y-17\right)}^2}{225}=1$

The dashed horizontal line represents the diameter of the open top of the fishbowl.  The maximum depth of the fishbowl, measured along the $y$-axis from the diameter of the open top to where the glass part of the bowl meets the base, is indicated by $d$  in the diagram.  All coordinates are expressed in centimetres, and for purposes of answering this question the thickness of the glass sides of the bowl may be regarded as negligible.

The owner of the company, Skodyn Pieseize, is extremely superstitious and is obsessed with the number 23. Therefore he insists that the capacity of the glass part of this new fishbowl must be exactly 23 litres.  Find the value of $d$ that satisfies this requirement.