Find the area bounded by the curve y = cos x + 1 and the lines x = 0 and x = a. Hence find the greatest area

Options

A. Area = sin a, Greatest area = 2 sq units
B. Area = cos a, Greatest area = 2 sq units
C. Area = sin a, Greatest area = 1 sq units
D. Area = cos a, Greatest area = 1 sq units

Answer

C

Explanation

Given y = cos x
Required area `=underset(0)overset(a)int(cosx)dx=intsinx]_(0)^(a)`
A = sin a
For the area to be greatest or least, `(dA)/(da)=0`
`implies cosa=0impliescosa=cos.(A)/(2)impliesa=(pi)/(2)`
Now, `(d^(2)A)/(da^(2))=-sina`
at `a=(pi)/(2),(d^(2)A)/(da^(2))=-sin.(pi)/(2)=-1lt0`
Therefore, the area is greatest at a `=(pi)/(2)`
And the greatest area = `sin.(pi)/(2)=1` sq units