 # Exam-Style Questions on Integration

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

### 1.

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.

### 2.

IB Standard

The acceleration, $$a$$ ms-2 , of an object at time $$t$$ seconds is given by

$$a=\frac1t+4sin3t, (t\ge1)$$

The object is at rest when $$t=1$$.

Find the velocity of the object when $$t=7$$.

### 3.

IB Applications and Interpretation

The following diagram shows part of the graph of:

$$f (x) = \frac{(9 + 4x) (6 - x)}{10}, x \in \mathbf R$$

The shaded region A is bounded by the x-axis, y-axis and the graph of $$f$$. (a) Write down an integral for the area of region A.

(b) Find the area of region A.

The three points A (0 , 0) , B (6 , 9) and C (p , 0) define the vertices of a triangle.

Find the value of p, the x-coordinate of C , such that the area of the triangle is equal to the area of region A.

### 4.

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.

### 5.

A-Level

(a) Express the algebraic fraction

$$\frac{6x^2 - 47x + 49}{(5-x)(1-2x)}$$

in the form

$$A + \frac{B}{5-x} + \frac{C}{1-2x}$$

where $$A$$, $$B$$ and $$C$$ are integers.

(b) Hence show that the following integral equates to 3.03 correct to three significant figures.

$$\int^{0.25}_0 \frac{6x^2 - 47x + 49}{(5-x)(1-2x)} dx$$

### 6.

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?

### 7.

IB Standard

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

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

### 8.

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.

### 9.

IB Standard

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

### 10.

A-Level

The diagram shows a sketch of the curve C with equation: $$y=2x^{\frac32}-4x+3$$

The point P with coordinates (4, 3) lies on C.

The line L is the tangent to C at the point P.

The region A, shown shaded, is bounded by the curve C, the line L and the y-axis.

Find the area of A making your method clear.

Note that solutions based entirely on graphical or numerical methods are not acceptable.)

### 11.

IB Standard

The following diagram shows the graph of $$f(x) = \cos(e^x) \; \text{for} \; 0 \le x \le 0.5$$. (a) Find the x-intercept of the graph of $$f(x)$$.

The region enclosed by the graph of $$f(x)$$, the y-axis and the x-axis is rotated 360° about the x-axis.

(b) Find the volume of the solid formed.

### 12.

IB Analysis and Approaches

Find $$f(x)$$ if:

$$f'(x) = \frac{12x}{\sqrt{3x^2+4}}$$

given that $$f(0) = 10$$

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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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