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INTEGRATION AND APPLICATIONS OF INTEGRATION

1. Integrate the following set of simple functions(i) dxxx

32 55 [4]

(ii) dxxx 2421 [4]

(iii)

dx

x

x2

72 [4]

2. Evaluate the following integrals:

(i)

45

x1

5e

5x

dx [4]

(ii) dxx1 [4]

(iii)

dxx

x44

3

1

4 [5]

3. Evaluate each of the following integrals

(i)

dxx

x2

3 [5]

(ii)

1

5ex

25

x

dx [4]

(iii) dxxx )5()4( 465 (Hint: Use substitution) [5]

4. Evaluate each of the following integrals

(i) dxxe x 564 24 [5]

(ii)

dxxx

45

5 [4]

(iii) dxxx )5()4( 465 (Hint: Use substitution) [5]

5. Evaluate each of the following integrals:(i)

dxx

xx

3

2 118 [4]

(ii) dxx 7 [4]

(iii) dxx

xex

1 [4]

6. Evaluate each of the following integrals

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(i)

dxx

xx2

12 [5]

(ii)

dxe

x

x3

5 [4]

(iii) dxxxx )54()5(364 (Hint: Use substitution) [5]

7. (a)Find the area under the curve 24 3 xy and the lines

x 1 and 2x

(b)Find the area above the x -axis bounded by the curve 236 xy and the lines 0x and 3x .(c) Find the area under the curve

y x 2 1 and the lines

x 1 and 3x [4]

(d)Find the area under the curve xxy 22 and the lines

x 1 and 4x

(e) Find the area under the curve y = 3x2 +2x +1 within the intervalx =1 andx = 1. [5](f) Find the area under curve

2

4

xxy , between the linesx = 2 andx = 4.

8. (a) The company also determined that its marginal profit in producing and sellingx units of the3GB flash drive is given by

xxP 06.0240)('

Find the original Profit function given that the company has fixed costs of $100,000.

(b) A company has determined that its marginal cost is given by

205.0)( xxC

Find the Cost function given that the company has fixed costs of $500. [4]

(c) A company has determined that its marginal cost is given by30025.0)( xxC

Find the Cost function given that the company has fixed costs of $2500. [4]

(d) A company has determined that its marginal cost is given by

75.0)( xxC

Find the Cost function given that the company has fixed costs of $1500. [5]

(e) A company has determined that its marginal revenue is given by

2001.0)( xxR

Find the Revenue function given that the company has fixed costs of $5000.