Area under and between Curves Worksheet

**Page 1**

1.

Find the area of the region bounded by $y$ = $x$^{2} + 2$x$ + 3, $x$-axis and the lines $x$ = - 1, $x$ = 1.

a. | $\frac{7}{3}$ | ||

b. | 4 | ||

c. | 2 | ||

d. | $\frac{20}{3}$ | ||

e. | $\frac{13}{3}$ |

The region R bounded by

The area of the region R =

[Definition.]

=

[Substitute

= [

= [

=

So, the area of the region =

Correct answer : (4)

2.

Find the area of the region bounded by $y$ = $x$^{2}, $x$- axis and the lines $x$ = 0, $x$ = 2.

a. | $\frac{4}{3}$ | ||

b. | 2 | ||

c. | 4 | ||

d. | 8 | ||

e. | $\frac{8}{3}$ |

The region R bounded by

The area of the region R =

=

[Substitute

= [

=

So, the area of the region =

Correct answer : (5)

3.

Find the area of the region bounded by the curves $y$ = $x$^{3} and $y$ = $x$.

a. | $\frac{3}{2}$ sq. units | ||

b. | 1 sq. unit | ||

c. | $\frac{1}{2}$ sq. units | ||

d. | 4 sq. units | ||

e. | 2 sq. units |

[Solve for

At

The points of intersection are (0, 0), (- 1, - 1) and (1, 1).

The region R bounded by the curves is the shaded region in the figure.

Area of the region = -

= [

= -

= 1 -

So, the area of the region is

Correct answer : (3)

4.

Find the area of the region bounded by the curves $y$ = $x$^{2} + 2 and $y$ = $x$ + 4.

a. | $\frac{9}{2}$ sq. units | ||

b. | $\frac{7}{6}$ sq. units | ||

c. | $\frac{10}{3}$ sq. units | ||

d. | 6 sq. units | ||

e. | 27 sq. units |

(

[Factor.]

[Solve for

At

The points of intersection are (2, 6) and (- 1, 3).

The region R bounded by the curves is the shaded region in the figure.

Area of the region =

[

=

= [-

= [-

= [

=

So, the area of the region is

Correct answer : (1)

5.

Find the area of the region bounded by the curves $y$ = $x$^{2} - 5$x$ and $y$ = 4 - 2$x$.

a. | $\frac{16}{3}$sq. units | ||

b. | $\frac{125}{6}$sq. units | ||

c. | $\frac{56}{3}$sq. units | ||

d. | $\frac{64}{3}$sq. units | ||

e. | $\frac{13}{6}$sq. units |

[Equate

(

[Solve for

At

The points of intersection are (- 1, 6) and (4, - 4).

The region R bounded by the curves is the shaded region in the figure.

The area of the region =

[Since (4 - 2

=

= [4

= [ 16 + 24 -

= [40 -

=

So, the area of the region =

Correct answer : (2)

6.

Find the area of the region bounded by the curves $y$ = $x$^{2} and $y$ = $x$^{4}.

a. | $\frac{1}{15}$sq. units | ||

b. | $\frac{16}{15}$sq. units | ||

c. | $\frac{2}{3}$sq. units | ||

d. | $\frac{4}{15}$sq. units | ||

e. | $\frac{2}{5}$sq. units |

[Equate

At

The points of intersection are (0, 0), (1, 1) and (-1, 1).

The region R bounded by the curves is the shaded region in the figure.

The area of the region =

[Since

= [

= [

=

So, the area of the given region =

Correct answer : (4)

7.

Find the area of the region bounded by the curves $y$ = 1 + 4$x$ - $x$^{2} and $y$ = 1 + $x$^{2}.

a. | $\frac{8}{3}$Sq.units | ||

b. | 8 Sq.units | ||

c. | $\frac{3}{8}$Sq.units | ||

d. | 3 Sq.units | ||

e. | $\frac{16}{3}$Sq.units |

1 + 4

[Equate

4

2

At

The points of intersection are (0, 1) and (2, 5).

The region R bounded by the curves is the shaded region in the figure.

The area of the region =

[Since (1 + 4

=

= [

= 8 -

So, the area of the region =

Correct answer : (1)