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Chapter 4: Q. 30 (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�01/2鈥�11+4x2dx

Short Answer

Expert verified

Ans: The exact value of鈭�01/2鈥�11+4x2dx=蟺8

Step by step solution

01

Step 1. Given information.

given expression,

鈭�01/2鈥�11+4x2dx

02

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

The exact value is calculated as shown below,

鈭�01/2鈥�11+4x2dx=鈭�01/2鈥�11+(2x)2dx=12tan鈭�1鈦�(2x)012=12tan鈭�1鈦�212鈭�tan鈭�1鈦�(0)=12蟺4鈭�0=蟺8

Therefore, the exact value is 蟺8.

03

Step 3. Check using a graph.

The required graph is:

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Most popular questions from this chapter

Use a sentence to describe what the notation鈭�k=2100k means. (Hint: Start with 鈥淭he sum of....鈥�)

Prove that 鈭�13(3x+4)dx=20in three different ways:

(a) algebraically, by calculating a limit of Riemann sums;

(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;

(c) with formulas, by using properties and formulas of definite integrals.

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. (Hint for Exercise 54: tanx=sinxcosx).

鈭�e3x-2e4xe2xdx.

Write each expression in Exercises 41鈥�43 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).

3鈭�k=225k2+2鈭�kk=224-鈭�k=0251

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