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QUESTION: 1

In the definite integral , the variable of integration is called

Solution:

QUESTION: 2

Express the shaded area in the form of an integral.

Solution:

As the curve goes from c to d and the equation is x = f(y)

So the shaded area is ∫(c to d)f(y)dy

QUESTION: 3

Evaluate as limit of sum

Solution:

∫(0 to 2)(x^{2} + x + 1)dx

= (0 to 2) [x^{3}/3 + x^{2}/2 + x]^{½}

= [8/3 + 4/2 + 2]

= 40/6

= 20/3

QUESTION: 4

Evaluate as limit of sum

Solution:

QUESTION: 5

The value of definite integral depends on

Solution:

QUESTION: 6

Find

Solution:

Using trigonometric identities, we have

cos^{2}x=cos^{2}x-sin^{2}x -(1) and cos^{2}x+sin^{2}x =1 -(2)

cos^{2}x=1-sin^{2}x , substituting this in equation (1) we get

cos^{2}x=1-sin^{2}x-sin^{2}x=1-2sin^{2}x

So,cos^{2}x=1-2sin^{2}x

2sin^{2}x=1-cos^{2}x

QUESTION: 7

Evaluate as limit of sum

Solution:

QUESTION: 8

The value of is:

Solution:

QUESTION: 9

Evaluate as limit of sum

Solution:

∫(0 to 4)3x dx

= [3x^{2}/2] (0 to 4)

[3(4)^{2}] / 2

= 24 sq unit

QUESTION: 10

The value of is:

Solution:

∫(0 to 3)1/[(3)^{2} - (x)^2]^{½}

∫1/[(a)^{2} - (x)^{2}] = sin^{-1}(x/a)

= [sin^{-1}(x/3)](0 to 3)

= sin^{-1}[3/3] - sin^{-1}[0/3]

= sin^{-1}[1]

= π/2

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