Definite integration

2021-12-26
The solid lies  between the Planes perpendicular to the x axis at X=-1amd X=1. Cross-sections perpendicular to the x axis between between these planes are circular disk with the diameters run from the semicircle y =-root(1-x*2)to the semi circle y=root(1-x*2).Find a formula for the area of cross section A(x)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

karton
Answered 2022-01-02 Author has 9103 answers

The bounds of the region are x -1 and x =1 so integrate the area function from -1 to 1 to find the volume.

\(\begin{array}{} V=\int^1_{-1}A(x)dx\\ V=\int^1_{-1} 4(1-x^2)dx\\ =4(x-\frac{1}{3}x^3)|^1_{-1}\\ =4(1-\frac{1}{3}(1)^3)-4(-1-\frac{1}{3}(-1)^3)\\ =4(1-\frac{1}{3})-4(-1+\frac{1}{3})\\ =4(\frac{2}{3})-4(-\frac{2}{3})\\ =\frac{8}{3}+\frac{8}{3}\\ =\frac{16}{3} \end{array}\)

Not exactly what you’re looking for?
Ask My Question
0

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-11-22
Evaluate the definite integral using integration by parts.
\(\displaystyle{\int_{{{0}}}^{{{2}}}}{z}{\left({z}-{2}\right)}^{{{4}}}{\left.{d}{z}\right.}\)
asked 2022-01-05
How to calculate following integration?
\(\displaystyle\int{\left(\sqrt{{{\tan{{x}}}}}+\sqrt{{{\cot{{x}}}}}\right)}{\left.{d}{x}\right.}\)
asked 2021-12-31
Demystify integration of \(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}\)
I've learned in my analysis class, that
\(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}={\ln{{\left({x}\right)}}}\)
I can live with that, and it's what I use when solving equations like that. But how can I solve this, without knowing that beforehand.
Assuming the standard rule for integration is
\(\displaystyle\int{x}^{{a}}{\left.{d}{x}\right.}={\frac{{{1}}}{{{a}+{1}}}}\cdot{x}^{{{a}+{1}}}+{C}\)
If I use that and apply this to \(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}\)
\(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}=\int{x}^{{-{1}}}{\left.{d}{x}\right.}\)
\(\displaystyle={\frac{{{1}}}{{-{1}+{1}}}}\cdot{x}^{{-{1}+{1}}}\)
\(\displaystyle={\frac{{{x}^{{0}}}}{{{0}}}}\)
Obviously, this doesn't work, as I get a division by 0. I don't really see, how I can end up with ln(x). There seems to be something very fundamental that I'm missing.
I study computer sciences, so, we usually omit things like in-depth math theory like that. We just learned that \(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}={\ln{{\left({x}\right)}}}\) and that's what we use.
asked 2021-12-28
Integration techniques. Use the methods introduced evaluate the following integrals.
\(\displaystyle\int{\frac{{{3}{x}}}{{\sqrt{{{x}+{4}}}}}}{\left.{d}{x}\right.}\)
asked 2021-12-19
For the integral state the number of the integration formula and the values of the constans a and b so that the formula fits the integral
\(\displaystyle\int{\frac{{{1}}}{{{x}^{{2}}{\left({4}{x}-{3}\right)}}}}{\left.{d}{x}\right.}\)
asked 2021-12-20
Using the technique of integration by parts evaluate the integral of the product of function \(\displaystyle{\left[{5}+{5}\right]}\)
a) \(\displaystyle{I}=\int{{\sin}^{{-{1}}}{x}}\ {\left.{d}{x}\right.}\)
b) \(\displaystyle{I}=\int{x}^{{{3}}}{\ln{{x}}}\ {\left.{d}{x}\right.}\)
asked 2021-12-16
Use the basic integration rules to find or evaluate the integral.
\(\displaystyle\int{x}{e}^{{{5}-{x}^{{{2}}}}}{\left.{d}{x}\right.}\)

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question
...