Exact value of the are of the region bounded by

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Homework Help Overview

The discussion revolves around finding the exact area of a region bounded by the curve x^3, the x-axis, and vertical lines at x=1 and x=4. Participants are also exploring a related question about the area under the curve e^x, bounded by the x-axis, y-axis, and the line x=1.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the process of finding the antiderivative of x^3 and evaluate it between the specified bounds. There is confusion regarding the correct limits of integration and the resulting area. Some participants question the accuracy of the provided answer of 3.75 and suggest that it may be a typo. Additionally, a new question is raised about finding the area under e^x, with participants expressing uncertainty about the setup involving both axes.

Discussion Status

The discussion is ongoing, with participants clarifying the bounds of integration and addressing discrepancies in the expected answer. There is no consensus yet on the correct area for the first problem, and the second question about e^x is still being explored.

Contextual Notes

Participants note that the answer provided in the book states "3 and 3/4 units^2," which has led to some frustration regarding its accuracy. The mention of the y-axis in the second question introduces additional complexity, as participants seek to understand how it relates to the area calculation.

kylebutler
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Question:
Find the exact value of the are of the region bounded by:
x^3, the x-axis and x=1 and x=4

Answer is 3.75

I tried finding the anti derivative so 1/4(x)^4, and therefore I've got 1/4(4)^4 - 1/4, which isn't the correct answer
 
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You are integrating x^3 for x from 0 to 1, right? So the antiderivative is, yes, x^4/4 and the result is 1^4/4-0^4/4=1/4. That's what you are doing yes? You've got a bunch of what I assume are 'dashes' in the post that look like 'minuses'. That looks ok to me, and I don't see how 3.75 could be correct.
 
I apologize, I meant x=1 and x=4 and I deleted all the dash (there was just one)
 
kylebutler said:
I apologize, I meant x=1 and x=4

That's different! So you did evaluate the antiderivative between 4 and 1. Then 4^4/4-1/4 is correct. And it's not 3.75. It's 63.75. A typo?
 
the answer in the book clearly states "3 and 3/4 units^2". If the answer is wrong than that's pretty awful, since it was the first problem. Other answers seem fine. Thanks for your help though.

If you don't mind another question I have is
Find the exact value of the are of the region bounded by:
e^x, the x-axis, the y-axis and x=1

I have no what to do if its bounded by both x and y axes
 
kylebutler said:
the answer in the book clearly states "3 and 3/4 units^2". If the answer is wrong than that's pretty awful, since it was the first problem. Other answers seem fine. Thanks for your help though.

If you don't mind another question I have is
Find the exact value of the are of the region bounded by:
e^x, the x-axis, the y-axis and x=1

I have no what to do if its bounded by both x and y axes

The 'y-axis' is x=0, right? That makes it pretty similar to the previous question.
 

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