Integration

WMA14Topic 4.64 questions

Substitution, by parts, partial fractions, volumes of revolution and differential equations.

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Original practice questions written in the style of the exam.

Question 15 marks
Use integration by parts to find the exact value of $\displaystyle\int_0^1 xe^{2x}\,dx$ .
Mark scheme
- M1 Uses parts with $u = x$ , $\dfrac{dv}{dx} = e^{2x}$ in the correct direction (1)
Question 24 marks
The region $R$ is bounded by the curve $y = 2\sqrt{x}$ , the -axis and the line .
Question 35 marks
Given that $y = 4$ when $x = 0$ , solve the differential equation

$\frac{dy}{dx} = 6x\sqrt{y}, \quad y > 0$

2x

A1

\tfrac{1}{2}xe^{2x} - \int \tfrac{1}{2}e^{2x}\,dx

(1)

A1

\tfrac{1}{2}xe^{2x} - \tfrac{1}{4}e^{2x}

(1)

M1 Substitutes limits 1 and 0 and subtracts (1)

A1

\dfrac{e^2 + 1}{4}

(1)

x

x

x = 3

$R$ is rotated through $2\pi$ radians about the $x$ -axis. Find the exact volume of the solid generated.

Mark scheme

dx

dy

=

6xy,y>

0

giving your answer in the form $y = f(x)$ .

Mark scheme

+

c