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Exam-Style Questions on IntegrationProblems 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 Analysis and Approaches |
Find:
$$ \int^{16}_9 \frac{6-2\sqrt{x}}{\sqrt{x}} dx $$by first writing the algebraic fraction in the form \(ax^b+c\).
4. | IB Analysis and Approaches |
(a) Find \(\int (4x+5) dx\).
(b) Given \(f'(x) = 4x+5\) find \(f(x)\) if \(f(3.4) = 10.12\).
5. | IB Analysis and Approaches |
Given that \( \frac{dy}{dx} = \sin(x + \frac{\pi}{3})\) and \(y = 5\) when \(x = \frac{8\pi}{3}\), find \(y\) in terms of \(x\).
6. | IB Analysis and Approaches |
The diagram below shows part of the graph of \(y = \dfrac{2x}{9-x^2}\)
The shaded region is bounded by the curve, the x-axis and the line \(x = c\)
The area of this region is \(\ln{2}\)
Find the value of \(c\).
7. | IB Analysis and Approaches |
The function \(f\) is defined by \(f(x) = 8 - 5 \sin{x} \), for \( x \ge 0 \).
The diagram shows part of the graph of \(y = f (x) \).
The shaded region is enclosed by the graph of \(y=f(x)\) and the x-axis for the first three periods of the function.
(a) Find the exact value of the x-coordinate of the right side of the shaded region.
(b) Show that the area of the shaded region is \( 48 \pi \).
A hemisphere has a total surface area in square centimetres equal to the shaded area in the previous diagram.
(c) Find the radius of the hemisphere.
8. | 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.
(c) 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.
9. | 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.
10. | 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.
$$ \int^{0.25}_0 \frac{6x^2 - 47x + 49}{(5-x)(1-2x)} dx $$11. | 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?
12. | IB Standard |
Find the value of \(a\) if \(\pi \lt a \lt 2\pi\) and:
$$ \int_\pi^a sin3x dx = -\frac13$$13. | 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.
14. | IB Standard |
Find \(f(x)\) if \(f'(x)=6 \sin2x\) and the graph of \(f(x)\) passes through the point \((\frac{\pi}{3},11)\).
15. | IB Analysis and Approaches |
The curve \( y=\sin(\sqrt{x}) \text{ where } x \ge 0 \) intersects the x axis at the points \(x_0, x_1, x_2, x_3, x_4, ... \) and \(x_0 = 0\).
(a) Find \(x_1\), the first x-intercept of the function to the right of the origin. Give your answer in terms of \(\pi\).
(b) Find an expression for the nth x-intercept of the curve, in terms of \(\pi\).
(c) By using an appropriate substitution, show that:
$$ \int \sin(\sqrt{x}) \; dx = 2\sin(\sqrt{x}) - 2 \sqrt{x} \cos(\sqrt{x})$$The area of the region bounded by the curve and the x-axis is denoted by \(R_n\) where:
$$ R_n = \int^{x_{n+1}}_{x_n} y \; dx$$
(d) Calculate the area of region \(R_n\) giving your answer in terms of \(\pi\).
(e) Hence, show that the areas of the regions bounded by the curve and the x-axis, form an arithmetic sequence.16. | IB Analysis and Approaches |
A function \( f \) is defined by \( f(x) = \frac{ke^{\frac{x}{2}}}{1 + e^x} \) where \( x \in \mathbb{R} \), \( x \geq 0 \) and \( k \in \mathbb{Z}^+ \).
The region enclosed by the graph of \( y = f(x) \), the \( x \)-axis, the \( y \)-axis and the line \( x = 3 \) is rotated \( 360^\circ \) about the \( x \)-axis to form a solid of revolution.
(a) Show that the volume of the solid formed is \( \pi k^2 \left( \frac{1}{2} - \frac{1}{1 + e^3} \right) \) cubic units.
(b) Find the minimum value of \(k\) such that the volume is at least 90 cubic units.
Sue wants to make a small bowl with a volume of 90 cm3 based on the result from parts (a) and (b).
Sue investigates how the cross-sectional radius of the bowl changes.
(c) By considering the graph of a suitable derivative of \( f \), find where the cross-sectional radius of the bowl is decreasing most rapidly.
(d) State the cross-sectional radius of the bowl at this point.
17. | 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.)
18. | 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.
19. | 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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