Chapter 13.2, Problem 32E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 33-36, evaluate each integral (a) with the Fundamental Theorem of Calculus and (b) with a graphing calculator (as a check). ∫ 2 0 x x 2 + 4 d x

(a)

To determine

To calculate: The value of the integral 02xx2+4dx using Fundamental Theorem of Calculus.

Explanation

Given Information:

The provided integral is 02xx2+4dx.

Formula used:

To calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

The value of the integral 1xdx=ln(x).

The value of the expression ln(ab)=lnalnb.

Fundamental Theorem of Calculus:

If f is a continuous function on a closed interval [a,b], then the definite integral of f exists and its value is abf(x)dx=F(b)F(a) where F is a function such that F(x)=f(x) for all x in [a,b].

Calculation:

Consider the integral 02xx2+4dx.

Recall that to calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

Thus, evaluate the integral.

Consider the expression (x2+4).

Differentiate with respect to x.

ddx(x2+4)=2xd(x2+4)=2xdx

Thus, d(x2+4)=2xdx.

Recall that if f is a continuous function on a closed interval [a,b], then the definite integral of f exists and its value is abf(x)dx=F(b)F(a) where F is a function such that F(x)=f(x) for all x in [a,b]

(b)

To determine

To calculate: The value of the integral 02xx2+4dx using a graphing calculator.

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