The value of ` int _ 3 ^ (4 ) sqrt((4- x ) (x - 3 )) dx ` is

Options

A. ` ( pi ) /( 16 ) `
B. ` ( pi ) /( 8 ) `
C. ` ( pi ) /( 4 ) `
D. ` ( pi ) /( 2 ) `

Answer

B

Explanation

Let `I = int_ 3^ 4 sqrt ((4 - x ) ( x - 3)) dx `
` = int _ 3 ^ 4 sqrt ( - x ^ 2 + 7x -12 ) dx `
` = int_ 3 ^ 4 sqrt ( - ( x ^ 2 - 7x + (49)/ ( 4 ) - ( 49)/( 4 )) - 12 ) dx `
= ` int_ 3 ^ 4 sqrt ( - ( x - ( 7 ) /( 2 ) ) ^ 2 + ( 49 ) /( 4) - 12 ) dx `
` = int _ 3 ^ 4 sqrt ( ( 1 ) / ( 4 ) - ( x - ( 7 ) / ( 2)) ^ 2 ) dx `
Let ` t = x - ( 7 ) /( 2 ) rArr dt = dx `
` therefore ` Upper limit = ` ( 1) / ( 2 ) ` and lower limit = - ` ( 1 ) / (2 ) `
` = int _ ( - 1//2 ) ^ ( 1//2 ) sqrt (( ( 1 )/( 2 ) ) ^ 2 - t ^ 2 ) dt `
` = 2 int _ 0 ^ ( 1//2 ) sqrt(( ( 1 )/( 2))^ 2 - t ^ 2 ) dt `
`= 2 [ ( t ) /(2) sqrt (( 1 ) /( 4) - t ^ 2 ) + ( 1 ) /( 8 ) sin ^( - 1 ) 2t] _ ( 0) ^( 1//2 ) `
` = 2 [0 + ( 1 ) /(8) xx ( pi ) / ( 2 )] `
` = ( pi )/( 8 ) `