Given the foll ouingaraph ot f(x)x -2x +3 or thea) interval3.C-1,3] , ApproximateP fix)dx by usingthe upper sums ivte neu. snadearea thatthetherepresentsupper sumsb)orer approximation oranIs the up per sumsunderapproximationforC fex) dx ? 131211106.f(x) = x - 2 a² + 39.5.212,3

Answered: Given the foll owing araph ot f(x)…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.7: Applied Problems

Problem 58E

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Question

Given the foll ouing
araph ot f(x)
x -2x +3 or the
a) interval
3.
C-1,3] , Approximate
P fix)dx by using
the upper sums ivte neu. snade
area that
the
the
represents
upper sums
b)
orer approximation or
an
Is the up per sums
under
approximation
for
C fex) dx ?

13
12
11
10
6.
f(x) = x - 2 a² + 3
9.
5.
21
2,
3

Expert Solution

Step 1

Given :

The graph of the function $f (x) = x^{3} - 2 x^{2} + 3$ in the interval [ -1 , 3 ] .

To determine :

(a) The approximate value of the integral $\int_{- 1}^{3} f (x) d x$ by using the upper sum s for n = 4. State the area that represents the upper sums.

(b) Is the upper sums an over a approximation or under approximation for $\int_{- 1}^{3} f (x) d x$ .

Step 2

The graph of the function $f (x) = x^{3} - 2 x^{2} + 3$ in the interval [ -1 , 3 ] and number of sub-intervals, n = 4.

Length of the interval

$\begin{array}{rcl} ∆ x & = & \frac{b - a}{n} \\ = & \frac{3 - (- 1)}{4} \\ = & \frac{3 + 1}{4} \\ = & \frac{4}{4} \\ = & 1 \end{array}$

Therefore, the sub - intervals are

( - 1 , 0 ) , ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 3 ) .

Step 3

(a)

The area of the the function $f (x) = x^{3} - 2 x^{2} + 3$ in the interval [ -1 , 3 ] using Riemann - upper sums is

$\begin{array}{rcl} \int_{- 1}^{3} f (x) d x & = & ∆ x [f (0) + f (1) + f (2) + f (3] \\ = & (1) [(0 + 0 + 3) + (1 - 2 + 3) + ((2^{3} - 2 {(2)}^{2} + 3) + ({(3^{3} - 2 (3)}^{2} + 3)] \\ = & 3 + 2 + (8 - 8 + 3) + (27 - 18 + 3) \\ = & 5 + 3 + 12 \\ = & 20 \end{array}$

Therefore, the area using upper sum approximation is 20 square units.

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