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Mandarinka [93]
3 years ago
14

Calculus I integral problem

Mathematics
1 answer:
mina [271]3 years ago
7 0

Answer:  -\frac{\sqrt{35}}{2}

In terms of typing this in on a keyboard, you could say -sqrt(35)/2

===================================================

Work Shown:

We'll use u-substitution here

Let u = x^2, so du/dx = 2x which rearranges to du = 2x*dx and we can say xdx = du/2

Because we are changing from x^2 to u, this means we need to change the limits of integration

If x = -2, then u = x^2 = (-2)^2 = 4

If x = 0, then u = x^2 = 0^2 = 0

Despite 4 being larger than 0, we still place 4 at the bottom limit and we'll just swap them later.

This means we have the following steps

\displaystyle \int_{-2}^{0}xg(x^2)\sqrt{7}dx\\\\\\\displaystyle \sqrt{7}\int_{-2}^{0}xg(x^2)dx\\\\\\\displaystyle \sqrt{7}\int_{-2}^{0}g(x^2)xdx\\\\\\\displaystyle \sqrt{7}\int_{4}^{0}g(u)\frac{du}{2}\\\\\\\displaystyle -\sqrt{7}\int_{0}^{4}g(u)\frac{du}{2}\\\\\\\displaystyle -\sqrt{7}*\frac{1}{2}\int_{0}^{4}g(u)du\\\\\\\displaystyle -\frac{\sqrt{7}}{2}\sqrt{5}\\\\\\\displaystyle -\frac{\sqrt{7}*\sqrt{5}}{2}\\\\\\\displaystyle -\frac{\sqrt{7*5}}{2}\\\\\\\displaystyle -\frac{\sqrt{35}}{2}\\\\\\

Notice when I swapped the limits of integration (4 and 0 to 0 and 4) I placed a negative sign out front.

The rule is \displaystyle \int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx

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Step-by-step explanation:

When margins are involved, the smallest area will be the one that has its dimensions in the same proportion as the margins. If x is the "multiplier", the dimensions of the printed area are ...

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_____

<em>Comment on margins</em>

It should be obvious that if both side margins are 4 cm, then the width of the poster is 8 cm more than the printed width. Similarly, the 6 cm top and bottom margins make the height of the poster 12 cm more than the height of the printed area.

_____

<em>Alternate solution</em>

Let w represent the width of the printed area. Then the printed height is 384/w, and the total poster area is ...

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Differentiating with respect to w gives ...

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<em>Generic solution</em>

If we let s and t represent the side and top margins, and we use "a" for the printed area, then the above equation becomes the symbolic equation ...

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For A' = 0, ...

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and the height is ...

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Then the ratio of width to height is ...

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  width/height = s/t . . . . . . the premise we started with, above

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