(a)

Trapezoidal Rule :

Let  be continuous on . The Trapezoidal Rule for approximating  is given by

, where , .

The function is  and .

Here ,  and .

.

.

Midpoint Rule :

Let  be continuous on . The Midpoint Rule for approximating  is given by

, where , .  

.

(b)

Error bounds :

Suppose for .If and are the errors in the Trapezoidal and Midpoint Rules, then and .

Consider .

Therefore, .

Substitute the values , , and in error bounds.

and

and

and . 

(c)

Consider .

Therefore, .

Trapezoidal error equation :

Accuracy to within means that the size of the error should be .

Midpoint error equation :

(a)

and .

(b)

and .

(c)

for .

for .