## 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 360^{o} 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

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 |

Consider the region where \(0 \lt x \lt 2\pi \) and \(\sin{x} \gt \cos{2x} \)

Find the area enclosed by the graphs of \(y=\sin{x} \) and \(y= \cos{2x} \)

## 8. | 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.

## 9. | 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.

## 10. | 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 360^{o} around the x-axis find the volume of the solid formed.

## 11. | 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.

$$ \int^{0.25}_0 \frac{6x^2 - 47x + 49}{(5-x)(1-2x)} dx $$## 12. | 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?

## 13. | IB Standard |

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

$$ \int_\pi^a sin3x dx = -\frac13$$## 14. | 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.

## 15. | IB Standard |

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

## 16. | 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.## 17. | 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 cm^{3} 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.

## 18. | 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.)

## 19. | 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.

## 20. | 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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