Definite integration

The graph of y = 4xsin 2x for 0 ≤ x ≤ π is shown below.

Show that the shaded area enclosed by the x-axis and the curve is kπ, where k is an integer to be found.

Fully justify your answer.

Use a suitable substitution to show that

can be written as

where a is a constant to be found.

Hence, or otherwise, show that

A graph has the equation

A student uses four rectangles to approximate the area under the graph between the lines x = 0 and x = 4

The rectangles are all the same width.

All the rectangles are drawn under the curve as shown in the diagram below.

The total area of the four rectangles is A

The student decides to improve their approximation by increasing the number of rectangles used.

Explain why the value of the student’s improved approximation will be greater than A, but less than

The region bounded by the curve

and the x-axis is shaded in the diagram below.

Use the trapezium rule with 5 ordinates to find an estimate for the area of the shaded region.

Give your answer correct to three significant figures.

Show that the exact area is given by

Fully justify your answer.

The shaded region, shown in the diagram below, is defined by

Identify which of the following gives the area of the shaded region.

Tick () one box.

Given that

use the quotient rule to show that

2 10 (b) The region enclosed by the curve y tan x and the horizontal line, which intersects the curve at x and x is shaded in the diagram below.

Show that the area of the shaded region is

Fully justify your answer.

The curve C is defined for t 0 by the parametric equations

2 2 3

C is shown in the diagram below.

Find the gradient of C at the point where it intersects the positive x-axis.

The area A enclosed between C and the x-axis is given by

Find the value of b.

Use the substitution y 4t² t³ to show that

2 3 4

Find the value of A.

Given that

find the exact value of a and the exact value of b.

Fully justify your answer.

Given that

deduce the value of

Circle your answer.

Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into n trapezia.

When n 4

State the number of ordinates that Caroline uses.

Calculate the area that Caroline should obtain using this method.

Give your answer correct to two decimal places.

Show that the exact area of A is

Fully justify your answer.

Explain what would happen to Caroline’s answer to part (a)(ii) as n

y e^x x x 16 (a) (sin cos)

Simplify your answer.

Hence, show that

where a is a rational number.

The areas of the finite regions bounded by the curve and the x-axis are denoted by A, A,..., A,...

Find the exact value of the area A₁

Show that

Given that

show that the exact value of the total area enclosed between the curve and the x-axis is

Show that

where p and q are rational numbers.

Find the first three terms, in ascending powers of x, of the binomial expansion

Hence, find the first three terms of the binomial expansion of

to seven decimal places.

Edward, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used.

Explain clearly whether Edward’s approximation will be an overestimate, an underestimate, or if it is impossible to tell.

Edward goes on to use the expansion from part (b) to find an approximation

Explain why Edward’s approximation is invalid.

Find the total shaded area.

Circle your answer.