Integration: Mathematics 9709 (Cambridge International AS & A Level)

Let $f(x) = 6x^2 + (4x + 1)^3$.

Which of the following is $\displaystyle\int f(x)\,dx$?

(a) Find $\displaystyle\int_1^4 \left(3\sqrt{x} + \frac{4}{x^2}\right) dx$. [4]

(b) Find $\displaystyle\int_{-1}^{2} (3x-1)^3\,dx$. [4]

A curve $C$ has equation $y = 8x - x^2$. A line $L$ has equation $y = 2x$.

(a) Find the $x$-coordinates of the points where $C$ and $L$ intersect. [2]

(b) Find the area of the finite region enclosed between $C$ and $L$. [6]

A curve $C$ passes through the point $(1, 9)$ and is such that $\frac{dy}{dx} = 12x^2 - \frac{6}{x^2} \quad (x>0).$

(a) Find the equation of the curve, giving $y$ in terms of $x$. [5]

(b) Find the value of $y$ when $x=3$. [3]

The region bounded by the curve $y = x+1$, the $x$-axis, and the lines $x=0$ and $x=3$ is rotated through $360°$ about the $x$-axis.

Find the volume of the solid formed, giving your answer as an exact multiple of $\pi$. [6]

The integrand of $\displaystyle\int_0^4 3x^{-1/2}\,dx$ is undefined at $x=0$, but this is a simple "improper" case where the integral still evaluates to a finite number.

What is the value of $\displaystyle\int_0^4 3x^{-1/2}\,dx$?

A curve has equation $y = x^2 - 6x + 5$.

(a) Find the two values of $x$ for which the curve crosses the $x$-axis. [2]

(b) The curve lies entirely below the $x$-axis between these two values of $x$. Find the area of the region enclosed between the curve and the $x$-axis. [5]

A region $R$ is bounded by the line $y = x+5$, the line $y = 2x+1$, and the $y$-axis, and lies between $x=0$ and the point where the two lines meet. The region $R$ does not touch the $x$-axis.

(a) Find the $x$-coordinate of the point where the two lines meet. [2]

(b) Find, as an exact multiple of $\pi$, the volume of the solid formed when $R$ is rotated through $360°$ about the $x$-axis. [6]

Which of the following is $\displaystyle\int (2x+3)^{-2}\,dx$?

Two curves have equations $y = x^2 - 2x$ and $y = 4x - x^2$.

(a) Find the $x$-coordinates of the points where the two curves intersect. [3]

(b) Find the area of the finite region enclosed between the two curves. [5]