Question

Find a horizontal line that divides the area between y=x^2 and y=9 into two equal parts.

Answer 1

I found the total area enclosed by the two curves (A=36) but how to a find a horizontal line that divides this area in half?

Answer 2

you'd want to set up an integral such that its result would yield half of this area

Answer 3

or in this case use symmetry. a parabola has equal area on either side of its line of symmetry. in this case the line of symmetry is x = 0

Answer 4

oh i think i misread your question. you need a horizontal line. sorry about that! in this case you'd want to integrate with respect to y

Answer 5

alternatively we could keep this with respect to x, and solve for the limits of integration:

\[2\int\limits_{0}^{a} (9 - x^{2}) dx = 18\]

Answer 6

\[\int\limits_{-a}^{a} (9 - x^{2}) dx = 18\]

This is equivalent (not sure why the "-a" doesn't show up as the lower limit, but that is what was typed.)

Answer 7

Ok I tried to solve but I could not get the right answer.
The answer in the book is \[\frac{ 9 }{ \sqrt[3]{4} }\]

Answer 8

I got two separate values, something like ~4.69 and ~1.08

Answer 9

@binarymimic

Answer 10

Ah, only one of those answers is in the domain of the problem. sqrt(9 - x^2)

Answer 11

ok but both of my answer were off

Answer 12

Sorry, 9 - x^2, not sqrt(9 - x^2)

Answer 13

\[18x-\frac{ 2x^3 }{ 3 }]^{a}_{0}=18\]

Answer 14

After that integrationi plugged into my calc and tried to find the intersection point to find a

Answer 15

\[\int\limits_{-1.04189}^{1.04189} (9 - x^{2}) dx \approx 18\]

Answer 16

should be a negative sign on the lower limit, but it doesn't show up anymore for some reason

Answer 17

answer in the book is \[\frac{ 9 }{ \sqrt[3]{4} }\]

Answer 18

The book says the horizontal line that divides the area into two equal parts is:
\[y = \frac{ 9 }{ \sqrt[3]{4} }\]

?

Answer 19

yes

Answer 20

sorry openstudy is glitchy again