Quick problem for math majors!?

Question

(a)Using a Riemann sum with left endpoints and n = 4, approximate the area under f(x) = x^ (4) + 2x + 1 from 3 to 9.

(b) Use an integral to find the exact value of this area.

Step by step please, especially part (b).

Thank you!

Answer 1

a)

Our range is from 3 to 9.  You have 4 divisions, or four rectangles.  Find the width of each one.

(9 - 3)/4 = 6 / 4 = 1.5 = width

The intervals are

(3, 4.5)(4.5, 6)(6, 7.5)(7.5, 9)

Evaluate each interval using the left endpoint.

Area = base * height

A = 1.5 * f(3) + 1.5 * f(4.5) + 1.5 * f(6) + 1.5 * f(7.5)

A = 1.5 (f(3) + f(4.5) + f(6) + f(7.5)

A = 1.5 (88 + 420.063 + 1309 + 3180.063)

A = 7495.688

b)

Area = integral of f(x) from 3 to 9

A = int [3, 9] (x^4 + 2x + 1) dx

Using power rule:

A = 1/5 x^5 + x^2 + x evaluated from 3 to 9

A = 1/5 (9^5) + (9^2) + 9 - (1/5 (3^5) + (3^2) + 3)

A = 11809.8 + 81 + 9 - 48.6 - 9 - 3

A = 11839.2

Answer 2

(a) The interval is broken up into 4 subintervals: 3-4.5, 4.5-6, 6-7.5, and 7.5-9.  Each has a width of 1.5.

Taking the left enpoints, it will evaluate as

(1.5)(f(3)+f(4.5)+f(6)+f(7.5))

Without a whizbang calculator, I'll leave the computation undone.

(b) Integrating x^4 + 2x + 1 gives

F(x) = (x^5)/5 + (2x^2)/2 + x = (x^5)/5 +x^2 + x

Given that, evaluate F(9) - F(3) to find the exact value of the area.

There should be a big difference between the two figures, since a 4th degree polynomial will get much bigger between x = 7.5 and x = 9