Areas and the trapezoidal rule

Calculate the exact area between a curve and an axis; find the area between two curves after determining intersection points; approximate areas numerically using the trapezoidal rule.

Worked examples

Area between two curves

Straightforward

Problem

Find the area of the region enclosed between

y = x + 2

and

y = x^2

Area when the curve dips below the $x$ -axis

Moderate

Problem

Find the total area enclosed between

y = x^2 - 4

and the

x

-axis.

The trapezoidal rule

Challenging

Problem

Use the trapezoidal rule with

n = 4

sub-intervals to approximate

\displaystyle\int_0^1 e^x\,dx

, and compare with the exact value.

Practise

Q1·Straightforward

Find the exact area between the curve

y = x^2 + 1

and the

x

-axis for

0 \leq x \leq 2

. Give your answer as a fraction.

Q2·Straightforward

Estimate

\displaystyle\int_0^2 x^2\,dx

using the trapezoidal rule with

n = 1

sub-interval.

Q3·Straightforward

Estimate

\displaystyle\int_0^2 x^2\,dx

using the trapezoidal rule with

n = 2

sub-intervals.

Q4·Straightforward

Find the exact area enclosed between

y = \cos x

and the

x

-axis for

0 \leq x \leq \dfrac{\pi}{2}

Q5·Straightforward

Find the exact area of the region enclosed between

y = x

and

y = x^2

for

0 \leq x \leq 1

. Give your answer as a fraction.

Q6·Moderate

Find the exact area of the region enclosed between the parabola

y = x^2

and the horizontal line

y = 4

. Give your answer as a fraction.

Q7·Moderate

Use the trapezoidal rule with

n = 4

sub-intervals to approximate

\displaystyle\int_1^3 \frac{1}{x}\,dx

. Give your answer correct to two decimal places.

Q8·Moderate

Find the exact area of the region enclosed between

y = x^3

and

y = x

for

0 \leq x \leq 1

. Give your answer as a fraction.

Q9·Moderate

Find the exact area under

y = e^x

from

x = -1

x = 1

, correct to two decimal places.

Q10·Moderate

Find the exact area of the region between

y = \sin x

and

y = \cos x

for

0 \leq x \leq \dfrac{\pi}{4}

, correct to two decimal places.

Q11·Challenging

Find the area of the region enclosed between the curves

y = x^2 - 2x

and

y = x

. Give your answer as a fraction.

Q12·Challenging

Use the trapezoidal rule with

n = 4

sub-intervals to estimate

\displaystyle\int_0^2 (x^2 + 1)\,dx

. By how much does this approximation exceed the exact value? Give your answer as a fraction.

Q13·Challenging

Find the exact area of the region enclosed between the curves

y = x^2 - 1

and

y = 2x + 2

. Give your answer as a fraction.

Open Math

Areas and the trapezoidal rule

Calculus · MAV-12-05

Name:

Date:

Q1Straightforward

Find the exact area between the curve

y = x^2 + 1

and the

x

-axis for

0 \leq x \leq 2

. Give your answer as a fraction.

Q2Straightforward

Estimate

\displaystyle\int_0^2 x^2\,dx

using the trapezoidal rule with

n = 1

sub-interval.

Q3Straightforward

Estimate

\displaystyle\int_0^2 x^2\,dx

using the trapezoidal rule with

n = 2

sub-intervals.

Q4Straightforward

Find the exact area enclosed between

y = \cos x

and the

x

-axis for

0 \leq x \leq \dfrac{\pi}{2}

Q5Straightforward

Find the exact area of the region enclosed between

y = x

and

y = x^2

for

0 \leq x \leq 1

. Give your answer as a fraction.

Q6Moderate

Find the exact area of the region enclosed between the parabola

y = x^2

and the horizontal line

y = 4

. Give your answer as a fraction.

Q7Moderate

Use the trapezoidal rule with

n = 4

sub-intervals to approximate

\displaystyle\int_1^3 \frac{1}{x}\,dx

. Give your answer correct to two decimal places.

Q8Moderate

Find the exact area of the region enclosed between

y = x^3

and

y = x

for

0 \leq x \leq 1

. Give your answer as a fraction.

Q9Moderate

Find the exact area under

y = e^x

from

x = -1

x = 1

, correct to two decimal places.

Q10Moderate

Find the exact area of the region between

y = \sin x

and

y = \cos x

for

0 \leq x \leq \dfrac{\pi}{4}

, correct to two decimal places.

Q11Challenging

Find the area of the region enclosed between the curves

y = x^2 - 2x

and

y = x

. Give your answer as a fraction.

Q12Challenging

Use the trapezoidal rule with

n = 4

sub-intervals to estimate

\displaystyle\int_0^2 (x^2 + 1)\,dx

. By how much does this approximation exceed the exact value? Give your answer as a fraction.

Q13Challenging

Find the exact area of the region enclosed between the curves

y = x^2 - 1

and

y = 2x + 2

. Give your answer as a fraction.

Worked solutions and answers at openmath.au/year-12/advanced/integral-calculus/areas-and-the-trapezoidal-rule

Areas and the trapezoidal rule

Worked examples

Area between two curves

Area when the curve dips below the xxx-axis

The trapezoidal rule

Practise

Areas and the trapezoidal rule

Area when the curve dips below the $x$ -axis