MATHEMATICS II

let us divide the interval

into four subintervals

and

Since the function f is an increasing function, it takes its smallest values at the left end of each interval. The length of each subinterval is

and

therefore

let us divide the interval

into four subintervals

and

Since the function f is an increasing function, it takes its smallest values at the left end of each interval. The length of each subinterval is

and

therefore

Thus lower sum of  corresponding to given partition is

Since

and since

Thus upper sum of  corresponding to the given partition is

We need to find the antiderivative F such that

One of these functions satisfying

If we invoke the Fundamental Theorem of Calculus  

Turning back to original variable , one of the antiderivative of the function  

Therefore the definite integral

on the interval

so the area A can be calculated by the definite integral

The antiderivative of

The correct answer is B.

The correct answer is A.

then we get

Returning back to the original variable x, we find that the antiderivative of

Thus the definite integral is

The correct answer is D.

The correct answer is C.

then we get

Then the correct answer is B.

We find the intersection points

For the interval

we have 

so the area A is evaluated as

The correct answer is C.

then we get

Returning back to the original variable x, we find that the antiderivative of

Thus the definite integral is

The correct answer is B.

The correct answer is D.

Let

and we get

and

With these results inserted in the integration by parts rule we obtain

The correct answer is E.