BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 5.4, Problem 22E
To determine

To find the value of g(π6) .

Expert Solution

Answer to Problem 22E

The value of g(π6) is 52 .

Explanation of Solution

Given information:

The function are f(x)=0sinx1+t2dt and g(y)=3yf(x)dx

Concept used:

The fundamental theorem of calculus, part 1 is defined by,

If f is continuous on [a,b] , then the function g defined by,

  g(x)=axf(t)dt

  axb is an antiderivative of f , that is g(x)=f(x) for a<x<b .

Calculation:

Using Part 1 of the Fundamental Theorem of Calculus, the derivative of the function is obtained as:

  f(x)=0sinx1+t2dt

By, evaluation theorem,

  ddx0sinx1+t2dt=F(sinx)F(x)=f(x)=1+sin2x

Now, for g(y)=3yf(x)dx ,

  g(y)=3y0sinx1+t2dtdx

Using Part 1 of the Fundamental Theorem of Calculus,

  g(y)=3yF(x)dx [since, 1+t2 is continuous and 0<x<b ]

Again, Using Part 1 of the Fundamental Theorem of Calculus,

  g(y)=F(y)g(y)=1+sin2yg(π6)=1+sin2(π6)g(π6)=1+(12)2g(π6)=1+14g(π6)=54g(π6)=52

Therefore,

The value of g(π6) is 52 .

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