Evaluate _1^2 ( 3 x^2 ) dx.

Write down the definite integral: _1^2 ( 3 x^2 ) dx Integrate using the reverse power rule: = [ x^3 ]_1^2 Substitute the limits (top minus bottom): = (8) - (1) Work out the value: = 7 Answer: 7

Evaluate _0^4 ( 2 x ) dx.

Write down the definite integral: _0^4 ( 2 x ) dx Integrate using the reverse power rule: = [ x^2 ]_0^4 Substitute the limits (top minus bottom): = (16) - (0) Work out the value: = 16 Answer: 16

Evaluate _1^2 ( 4 x^3 ) dx.

Write down the definite integral: _1^2 ( 4 x^3 ) dx Integrate using the reverse power rule: = [ x^4 ]_1^2 Substitute the limits (top minus bottom): = (16) - (1) Work out the value: = 15 Answer: 15

Evaluate _0^3 ( 6 x + 1 ) dx.

Write down the definite integral: _0^3 ( 6 x + 1 ) dx Integrate using the reverse power rule: = [ 3 x^2 + x ]_0^3 Substitute the limits (top minus bottom): = (30) - (0) Work out the value: = 30 Answer: 30

Evaluate _0^3 ( x^2 + 2 ) dx.

Write down the definite integral: _0^3 ( x^2 + 2 ) dx Integrate using the reverse power rule: = [ x^33 + 2 x ]_0^3 Substitute the limits (top minus bottom): = (15) - (0) Work out the value: = 15 Answer: 15

Definite integration and areas Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Definite integration and areas questions. See exactly how to solve problems on definite-integral, power-rule, linear, constant.

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A-Level70 questionsStep-by-step solutions

Question 1

2 markseasy

Evaluate

\displaystyle \int_{1}^{2} \left( 3 x^{2} \right) \, dx

Worked solution

Write down the definite integral
$\int_{1}^{2} \left( 3 x^{2} \right) \, dx$
Identify the function to integrate and note the lower limit (bottom number) and upper limit (top number).
Integrate using the reverse power rule
$= \left[ x^{3} \right]_{1}^{2}$
Add one to each power and divide by the new power, then keep the limits attached to the square bracket.
Substitute the limits (top minus bottom)
$= \left(8\right) - \left(1\right)$
Put in the upper limit first, then subtract the value you get from the lower limit.
Work out the value
$= 7$
Finish the arithmetic to get the single number.

Answer

7

Question 2

2 markseasy

Evaluate

\displaystyle \int_{0}^{4} \left( 2 x \right) \, dx

Worked solution

Write down the definite integral
$\int_{0}^{4} \left( 2 x \right) \, dx$
Identify the function to integrate and note the lower limit (bottom number) and upper limit (top number).
Integrate using the reverse power rule
$= \left[ x^{2} \right]_{0}^{4}$
Add one to each power and divide by the new power, then keep the limits attached to the square bracket.
Substitute the limits (top minus bottom)
$= \left(16\right) - \left(0\right)$
Put in the upper limit first, then subtract the value you get from the lower limit.
Work out the value
$= 16$
Finish the arithmetic to get the single number.

Answer

16

Question 3

2 markseasy

Evaluate

\displaystyle \int_{1}^{2} \left( 4 x^{3} \right) \, dx

Worked solution

Write down the definite integral
$\int_{1}^{2} \left( 4 x^{3} \right) \, dx$
Identify the function to integrate and note the lower limit (bottom number) and upper limit (top number).
Integrate using the reverse power rule
$= \left[ x^{4} \right]_{1}^{2}$
Add one to each power and divide by the new power, then keep the limits attached to the square bracket.
Substitute the limits (top minus bottom)
$= \left(16\right) - \left(1\right)$
Put in the upper limit first, then subtract the value you get from the lower limit.
Work out the value
$= 15$
Finish the arithmetic to get the single number.

Answer

15

Question 4

2 markseasy

Evaluate

\displaystyle \int_{0}^{3} \left( 6 x + 1 \right) \, dx

Worked solution

Write down the definite integral
$\int_{0}^{3} \left( 6 x + 1 \right) \, dx$
Identify the function to integrate and note the lower limit (bottom number) and upper limit (top number).
Integrate using the reverse power rule
$= \left[ 3 x^{2} + x \right]_{0}^{3}$
Add one to each power and divide by the new power, then keep the limits attached to the square bracket.
Substitute the limits (top minus bottom)
$= \left(30\right) - \left(0\right)$
Put in the upper limit first, then subtract the value you get from the lower limit.
Work out the value
$= 30$
Finish the arithmetic to get the single number.

Answer

30

Question 5

2 markseasy

Evaluate

\displaystyle \int_{0}^{3} \left( x^{2} + 2 \right) \, dx

Worked solution

Write down the definite integral
$\int_{0}^{3} \left( x^{2} + 2 \right) \, dx$
Identify the function to integrate and note the lower limit (bottom number) and upper limit (top number).
Integrate using the reverse power rule
$= \left[ \frac{x^{3}}{3} + 2 x \right]_{0}^{3}$
Add one to each power and divide by the new power, then keep the limits attached to the square bracket.
Substitute the limits (top minus bottom)
$= \left(15\right) - \left(0\right)$
Put in the upper limit first, then subtract the value you get from the lower limit.
Work out the value
$= 15$
Finish the arithmetic to get the single number.

Answer

15

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