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Chapter 4: Q. 32 (page 373)

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

鈭�24鈥�3e2x鈭�4dx

Short Answer

Expert verified

Ans: The exact value of,鈭�24鈥�3e2x鈭�4dx=32e4

Step by step solution

01

Step 1. Given information.

given,

鈭�24鈥�3e2x鈭�4dx

02

Step 2. The objective is to determine the exact value of the definite integral.  

The exact value is calculated as shown below,

鈭�24鈥�3e2x鈭�4dx=3鈭�24鈥�e2x鈭�4dx=312e2x鈭�324=32e2(4)鈭�3鈭�e2(2)鈭�3=32e5鈭�e1=32e4

Therefore, the exact value is,32e4

03

Step 3. Check

The required graph is,

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Most popular questions from this chapter

Sum and constant-multiple rules: State the sum and constant-multiple rules for (a) derivatives and (b) limits.

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating .

鈭�23x+1dx

Consider the region between f and g on [0, 4] as in the

graph next at the left. (a) Draw the rectangles of the left-

sum approximation for the area of this region, with n = 8.

Then (b) express the area of the region with definite

integrals that do not involve absolute values.

Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)

(a) The signed area between the graph of f(x) = cos x and the x-axis on [鈭捪�, 蟺].

(b) The average value of f(x) = cos x on [0, 2蟺].

(c) The area of the region between the graphs of f(x) =4鈭�x2andg(x)=鈭�4鈭�x2on[鈭�2,2].

Write out all the integration formulas and rules that we know at this point.
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