What's the closest integer approximation?

Question

Let R be the region enclosed by the graph of y=sin e^x^2, the x-axis, and the lines x=-1 and x=1. What's the closest integer approximation of the area of R?

 

Answer: 2

 

Since it's x-axis, don't you set up a integral with dx at the end? How do I take that integral? Show your work.

Answer 1

area = 2){0, 1}sin(e^x^2)) dx = 1.772 ~ 2 (calculated by TI-84)

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Estimation without using a graphing calculator:

f(x) = sin(e^x^2))

f'(x) = cos(e^x^2))e^(x^2) (2x) on [0, 1]

Solve f'(x) = 0 for x,e^x^2 = pi/2

x = sqrt[ln(pi/2)]

So, fmin = f(0) = sin(1), and fmax = f(sqrt[ln(pi/2)]) = sin(pi/2) = 1

2sin(1) < 2){0, 1}sin(e^x^2)) dx < 2*1

1.683 < 2){0, 1}sin(e^x^2)) dx < 2

Answer: 2

 

Answer 2

The integral  sin(e^x^2)) probably cannot be done in closed form.  However, the maximum of the function is 1 and the function is slowly varying on the interval -1 to 1  with an average value greater than 0.75.  Consequently, the integral  on the interval -1 to 1 will be greater than 1.5 and less than 2.   This means that closest integer approximation is 2.