definite integral

Question

1. Properties of the definite integral:

a) If ∫ 2 to 9 f(x)dx=23 and ∫2 to 4 f(x)dx=5 , find ∫ 4 to 9 f(x)dx

b) If ∫ 1 to 10 f(x)dx=11 and ∫3 to 10 f(x)dx=−5 , find ∫1 to 3 f(x)dx

c) If ∫ 0 to 8 f(x)dx=35/2 and ∫5 to 8 f(x)dx=3 , find ∫5 to 0  2f(x)dx

d) ∫ 0 to 2 1+sqrt(4−x^2) dx (Hint: interpret it in term of area)

 

2. True or False. Hint: think symmetry if appropriate

a) ∫ -1 to 1 (sinx)^(1/3) dx=0

b) ∫-2 to 2  (x^2 cosx)/( 1+x^2) dx=2∫ 0 to 2 (x^2cosx)/( 1+x^2) dx20

c) ∫a to b f(x)/g(x) dx =( ∫a to b f(x)dx)/ ( ∫a to b g(x)dx)  , if ∫ a to b g(x)dx ≠0

d) ∫ a to b [f(x)]^2dx=[ ∫ a to b f(x)dx]^2  provided the integral exist

Answer 1

1.(a)

Step 1:

The integrals are and .

The integral is .

Apply Additive Interval Property .

Substitute and .

.

Solution:

.

Answer 2

1.(b)

Step 1:

The integrals are and .

The integral is .

Apply Additive interval property .

Substitute and .

.

Solution:

.

Answer 3

1.(c)

Step 1:

The integrals are and .

The integral is .

Find .

Apply Additive Interval Property .

Substitute and .

.

.

Step 2:

Find .

Apply Special Definite Integrals:.

Substitute in .

.

.

Solution:

.

Answer 4

(2)

Step 1:

(a)

The integral is .

Symmetry property of integrals:

If is an even function then ,

and if is an odd function then , .

Here the integrand function is  .

Replace in the above function.

Thus, the function is odd function.

Therefore, .

The statement is true.

Solution:

The statement is true.

Answer 5

(2)

Step 1:

(b)

The integral is .

Symmetry property of integrals:

If is an even function then , .

and if is an odd function then , .

Here, the integrand function is .

Replace in the above function.

Thus, the function is an even function.

Therefore, by the symmetry properties of integrals,

.

The statement is true.

Solution:

The statement is true.

Answer 6

(2)

Step 1: 

(c)

The integral is .

Rewrite the integral as .

Determine the integral by using integration byparts.

Integration by parts: .

.

Therefore, .

 

The statement is false.

Solution:

The statement is false.

Answer 7

(2)

Step 1:

(d)

The integral is .

Power rule of integration:.

.

Therefore, .

The statement is false.

Solution:

The statement is false.

Answer 8

1.(d)

Step 1:

The integral is .

Area .

Apply properties of definite integrals:.

.

Step 2:

Consider .

Substitute .

.

Substitute  in the above expression.

.

.

Substitute in .

.

Solution:

Area is square-units.