# Exam-Style Questions on Integration

## Problems on Integration adapted from questions set in previous exams.

### 1.

IB Standard

Make a sketch of a graph showing the velocity (in $$ms^{-1}$$) against time of a particle travelling for six seconds according to the equation:

$$v=e^{\sin t}-1$$

(a) Find the point at which the graph crosses the $$t$$ axis.

(b) How far does the particle travel during these first six seconds?

### 2.

IB Standard

Consider the graph of the function $$f(x)=x^2+2$$.

(a) Find the area between the graph of $$f$$ and the x-axis for $$2\le x \le 3$$.

(b) If the area described above is rotated 360o around the x-axis find the volume of the solid formed.

### 3.

IB Standard

Find the value of $$a$$ if $$\pi \lt a \lt 2\pi$$ and:

$$\int_\pi^a sin3x dx = -\frac13$$

### 4.

IB Standard

This graph represents the function $$f:x\to a \cos x, a\in \mathbf N$$

(a) Find the value of $$a$$.

(b) Find the area of the shaded region.

### 5.

IB Standard

Find $$f(x)$$ if $$f'(x)=6 \sin2x$$ and the graph of $$f(x)$$ passes through the point $$(\frac{\pi}{3},11)$$.

### 6.

IB Standard

The graph of $$f(x)=8-x^2$$ crosses the x-axis at the points A and B.

(a) Find the x-coordinate of A and of B.

(b) The region enclosed by the graph of $$f$$ and the x-axis is revolved 360o about the x-axis. Find the volume of the solid formed.

### 7.

IB Studies

The cross-section of a fish pond is drawn on a set of axes shown below. The edge is modelled by $$y=ax^2+c$$ and the cross section is the same for the whole of its length. The curve touches the x-axis at the origin.

Point A has coordinates (-9,5.4) and point B has coordinates (9,5.4).

(a) Find the value of $$c$$.

(b) Find the value of $$a$$.

(c) Hence write down the equation of the quadratic function which models the edge of the fish pond.

(d) Calculate the value of $$y$$ when $$x$$=7.2m.

(e) State what the value of $$x$$ and the value of $$y$$ represent for this fish pond.

(f) Find the value of $$x$$ when the height of water in the pond is 2.7m.

The pond is filled to a maximum depth of 2.7m and the the cross-sectional area of the water is 22.9m2. The pond has a length of 8m.

(g) Calculate the volume of water in the pond.

### 8.

IB Standard

Let $$f(x)=\frac{2x}{x^2+3}$$

(a) Use the quotient rule to show that $$f'(x)=\frac{6-2x^2}{(x^2+3)^2}$$.

(b) Find $$\int \frac{2x}{x^2+3}dx$$.

(c) The area enclosed by the graph of $$f(x)$$, the x-axis and the lines $$x=\sqrt3$$ and $$x=n$$ has an area of $$\ln14$$. Find the value of $$n$$.

The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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