Problem involving a definite integral

In summary: So I think it would make sense to divide the interval [0, 1] into two parts, and calculate the areas of the two rectangular areas under the graph of ##y = e^{x^2}##. Then you could use the lower bound to get an upper bound on the integral.So you're thinking of doing a Riemann sum?Yes.Yes. then from there on how do I proceed? Options B and C get ruled out.Can you fill in for the question mark?In summary, the integral e^(x^2)dx from 0 to 1 appears to be less than the integral e^x from 0 to 1, so a is clearly greater than 1.
  • #1
ubergewehr273
142
5

Homework Statement


Refer the image.

Homework Equations


Integral (e^x)dx from 0 to 1=e-1.

The Attempt at a Solution


Refer the other image. The graph of e^(x^2) increases more slowly than e^x till x=1.
So 'a' is clearly greater than 1. Is this right?
 

Attachments

  • Screenshot-2018-5-8  sol ps - NEST-13-A pdf.png
    Screenshot-2018-5-8 sol ps - NEST-13-A pdf.png
    6.8 KB · Views: 510
  • 20180508_224300.jpg
    20180508_224300.jpg
    19.1 KB · Views: 466
Physics news on Phys.org
  • #2
Thread moved. Please post problems involving integrals in the Calculus & Beyond section, not in the Precalculus section.

ubergewehr273 said:
The graph of e^(x^2) increases more slowly than e^x till x=1.
What does ##e^x## have to do with anything? The integrand is ##e^{x^2}##.
 
  • #3
Mark44 said:
What does ##e^x## have to do with anything? The integrand is ##e^{x^2}##.
So that ##\int_0^1 e^{x^2}\,dx## and ##\int_0^1 e^x\,dx## can be compared.
Clearly ##\int_0^1 e^{x^2}\,dx## < ##\int_0^1 e^x\,dx##
 
  • #4
ubergewehr273 said:
So that ##\int_0^1 e^{x^2}\,dx## and ##\int_0^1 e^x\,dx## can be compared.
Clearly ##\int_0^1 e^{x^2}\,dx## < ##\int_0^1 e^x\,dx##
Sure, they can be compared.
Relative to this problem, you have ##e - 1 = a\int_0^1 e^{x^2}dx < a\int_0^1 e^x dx =##?
Can you fill in for the question mark?
What does this say about a?
Do you still believe that a > 1?
 
Last edited:
  • #5
ubergewehr273 said:
So that ##\int_0^1 e^{x^2}\,dx## and ##\int_0^1 e^x\,dx## can be compared.
Clearly ##\int_0^1 e^{x^2}\,dx## < ##\int_0^1 e^x\,dx##
##a=\frac{e-1} {\int_0^1 e^{x^2}\,dx}##
And from the above quoted equation,
##a=\frac{e-1} {<e-1}##
Hence ##a>1##. Is this correct?
 
  • #6
ubergewehr273 said:
##a=\frac{e-1} {\int_0^1 e^{x^2}\,dx}##
And from the above quoted equation,
##a=\frac{e-1} {<e-1}##
Hence ##a>1##. Is this correct?
Yes.
 
  • #7
Mark44 said:
Yes.
Then from there on how do I proceed? Options B and C get ruled out.
 
  • #8
You have an upper bound on ##\int_0^1 e^{x^2}dx##, namely ##\int_0^1 e^x = e - 1##, so now you need a lower bound on that integral.
What about a Riemann sum? You could get a lower bound by dividing the interval [0, 1] into two parts, and calculating the areas of the two rectangular areas under the graph of ##y = e^{x^2}##.

Here's what I have in mind -- it's a pretty crude sketch, but maybe it gets my idea across. The curve is supposed to be the graph of ##y = e^{x^2}##.
graph.png
 

Attachments

  • graph.png
    graph.png
    953 bytes · Views: 401
Last edited:

1. What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve on a graph. It is usually denoted by ∫ (the integral symbol) and has two limits, a lower and an upper bound, which indicates the range over which the area is being calculated.

2. What types of problems involve definite integrals?

Definite integrals can be used to solve a variety of problems, such as finding the area under a curve, calculating displacement and velocity, determining the volume of a solid, and evaluating the average value of a function. They are also commonly used in physics, engineering, and economics.

3. How do you solve a problem involving a definite integral?

To solve a problem involving a definite integral, you first need to identify the function that represents the curve and the limits of integration. Then, you can use techniques such as the Riemann sum, the fundamental theorem of calculus, or integration by substitution to evaluate the integral and find the desired solution.

4. What are the common mistakes when solving a problem involving a definite integral?

Some common mistakes when solving problems involving definite integrals include forgetting to include the limits of integration, using incorrect integration techniques, and making errors in algebraic manipulations. It is also important to pay attention to the units and interpret the final answer correctly.

5. Can definite integrals be solved using technology?

Yes, definite integrals can be solved using technology such as graphing calculators or computational software. These tools can help with complex calculations and provide visual representations of the area under a curve. However, it is still important to understand the concepts and techniques involved in solving definite integrals manually.

Similar threads

  • Calculus and Beyond Homework Help
Replies
7
Views
606
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
431
  • Calculus and Beyond Homework Help
Replies
8
Views
692
  • Calculus and Beyond Homework Help
Replies
10
Views
204
  • Calculus and Beyond Homework Help
Replies
4
Views
866
  • Calculus and Beyond Homework Help
Replies
2
Views
782
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
783
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
Back
Top