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11-7 Area Between Curves - Printable Mathematics Worksheets year 12 - Wayground

11-7 Area Between Curves

18 questions

12th - 12th Grade

Mathematics

This Grade 12 mathematics worksheet gives students 18 multiple-choice questions on finding the area between curves. Students work with regions bounded by lines, parabolas, and trigonometric graphs, including examples such as a line and a quadratic, two parabolas, and sine-based curves over a stated interval. Several items also ask students to evaluate integrals that support area calculations. The worksheet emphasizes setting up the correct bounded region, identifying where graphs intersect, determining which function is above the other, and selecting the resulting area from several choices. It is useful for calculus practice, independent review, homework, or a quick formative assessment after instruction on definite integrals and area between curves. This free printable worksheet includes a complete answer key for efficient checking and feedback.

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Worksheets

11-7 Area Between Curves

Total questions: 18

Name
Class
Date
1.

Find the area of the region enclosed by the lines and curves.

y = 3x + 4 and y = x² + 4

a)

9

b)

93/2

c)

9/2

d)

18

2.
a)
A
b)
B
c)
C
d)
D
3.
a)
A
b)
B
c)
C
d)
D
4.

Find the area of the region bounded by the equations y = x2 + 1 and y = 5.

a)

32/3

b)

64

c)

32

d)

64/3

e)

16

5.
a)
A
b)
B
c)
C
d)
D
6.
a)
A
b)
B
c)
C
d)
D
7.
a)
A
b)
B
c)
C
d)
D
8.

Comping this because one of these is not a function

a)

A

b)

B

c)

C*****

d)

D

9.
a)
A
b)
B
c)
C
d)
D
10.

Find the area of the region enclosed by the lines and curves.

y = -4sin(x) y = sin(2x) 0 ≤ x ≤ Π

a)

8

b)

1/2

c)

4

d)

16

11.

Find the area of the region bounded by the parabolas y = 2x - x2 and y = x2.

a)

3

b)

1

c)

-3

d)

1/3

12.

Evaluate the integral.

a)

−54cos⁡(2x2) + c-\frac{5}{4}\cos\left(2x^2\right)\ +\ c

b)

54cos⁡(2x2) + c\frac{5}{4}\cos\left(2x^2\right)\ +\ c

c)

−54cos⁡(2x2)-\frac{5}{4}\cos\left(2x^2\right)

d)

can't be integrated

13.

Find the area of the region enclosed by the lines and curves.

y = 3x + 4 and y = x² + 4

a)

9

b)

93/2

c)

9/2

d)

18

14.

Evaluate the integral.
∫01x2e2x3−1 dx\int_0^1x^2e^{2x^3-1\ }dx  

a)

16(e − 1e)\frac{1}{6}\left(e\ -\ \frac{1}{e}\right)  

b)

6(e − 1e)6\left(e\ -\ \frac{1}{e}\right)  

c)

16(e −1)\frac{1}{6}\left(e\ -1\right)  

d)

6(e−1)6\left(e-1\right)  

15.
Find the area of the region enclosed by the lines and curves.
y = -4sin(x)   y = sin(2x)    0 ≤ x ≤ Π 
a)
8
b)
1/2
c)
4
d)
16
16.

What is the area of the region between the graphs

f(x)=3x−x2f\left(x\right)=3x-x^2  and x−axisx-axis from x=0x=0 and x=2x=2  

a)

103\frac{10}{3}  

b)

33

c)

44  

d)

113\frac{11}{3}  

17.

What is the area of the region between the graps y=x2−1y=x^2-1 and line y=−x−1y=-x-1  

a)

14\frac{1}{4}  

b)

15\frac{1}{5}  

c)

16\frac{1}{6}  

d)

17\frac{1}{7}  

18.

Find the area of the region bounded by the parabolas y = 2x - x2 and y = x2.

a)

3

b)

1

c)

-3

d)

1/3

Answer Key

11-7 Area Between Curves

Total questions: 18

1.
c)

9/2

2.
a) A
3.
c) C
4.
a)

32/3

5.
d) D
6.
a) A
7.
d) D
8.
c)

C*****

9.
c) C
10.
a)

8

11.
d)

1/3

12.
a)

−54cos⁡(2x2) + c-\frac{5}{4}\cos\left(2x^2\right)\ +\ c

13.
c)

9/2

14.
a)

16(e − 1e)\frac{1}{6}\left(e\ -\ \frac{1}{e}\right)  

15.
a) 8
16.
a)

103\frac{10}{3}  

17.
c)

16\frac{1}{6}  

18.
d)

1/3

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FAQs

What does the 11-7 Area Between Curves worksheet cover?

This Grade 12 worksheet uses 18 multiple-choice questions to practise finding the area of regions bounded by lines, parabolas, and trigonometric curves. Students must identify relevant intersections or intervals, compare the graphs, set up an appropriate integral, and evaluate it to choose the area.

How can I use this worksheet to teach area between curves?

Use the line-and-parabola, parabola-and-parabola, and trigonometric examples to model the same process across different graph types: identify the bounds, determine the upper and lower functions, and integrate their difference. Before students begin the multiple-choice questions, ask them to sketch one region and explain why its area must be nonnegative.

What mistakes should I watch for when students complete this worksheet?

Students may reverse the upper and lower functions, use incorrect intersection points, or calculate a signed integral instead of the geometric area. Watch especially for errors on the trigonometric region over 0 ≤ x ≤ π, where students must respect the stated interval, and on the parabola questions, where a negative choice can reflect a setup or evaluation mistake rather than an area.

How do I assign and grade this worksheet?

The worksheet is available as a printable PDF and as a digital quiz on Wayground, and it includes a complete answer key. For printable submissions, scan student work with the Wayground for Teachers app to support faster grading.

How can I differentiate this area-between-curves worksheet for my students?

In the worksheet’s Advanced Settings, adjust font spacing or choose a larger font size to make the multiple-choice area and integral questions easier to read. You can also enable a dyslexia-friendly font or translate the worksheet into another language when those file-level supports are useful.

Where can I find more worksheets like this on Wayground?

Wayground offers a broad collection of free printable worksheets and practice problems across subjects, with downloadable PDFs and answer keys to support essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

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