Volumes of revolution are so fun
lets say you have 2 functions f(x) and g(x) where they intersect at c and d where c<d, and f(x) is on top of g(x) (like for a value r where c<r<d, f(r)>g(r)), the volume is
![\pi \int\limits^d_c {f(x)^2-g(x)^2} \, dx](https://tex.z-dn.net/?f=%20%5Cpi%20%5Cint%5Climits%5Ed_c%20%7Bf%28x%29%5E2-g%28x%29%5E2%7D%20%5C%2C%20dx%20)
ok, so find the intersection points of
![y=9x^3](https://tex.z-dn.net/?f=y%3D9x%5E3)
and y=9
9=9x^3
1=x^3
at x=1
so then the area will be from x=0 to x=1
which ones's higher?
try x=0.5
at x=0.5, 9(0.5)^3<9
so the y=9 is on top
therfor
limits are from 0 to 1
higher one is y=9
bottom function is y=9x^3
so the volume is
![\pi \int\limits^1_0 {(9)^2-(9x^3)^2} \, dx =](https://tex.z-dn.net/?f=%20%5Cpi%20%5Cint%5Climits%5E1_0%20%7B%289%29%5E2-%289x%5E3%29%5E2%7D%20%5C%2C%20dx%20%3D)
![\pi \int\limits^1_0 {81-81x^6} \, dx =](https://tex.z-dn.net/?f=%20%5Cpi%20%5Cint%5Climits%5E1_0%20%7B81-81x%5E6%7D%20%5C%2C%20dx%20%3D)
we can factor out the 81
![\pi \int\limits^1_0 {81(1-x^6)} \, dx =](https://tex.z-dn.net/?f=%20%5Cpi%20%5Cint%5Climits%5E1_0%20%7B81%281-x%5E6%29%7D%20%5C%2C%20dx%20%3D)
factor out the constant
![81 \pi \int\limits^1_0 {1-x^6} \, dx =](https://tex.z-dn.net/?f=%2081%20%5Cpi%20%5Cint%5Climits%5E1_0%20%7B1-x%5E6%7D%20%5C%2C%20dx%20%3D)
integrate
![81 \pi [x-\frac{1}{7}x^7]^1_0=](https://tex.z-dn.net/?f=81%20%5Cpi%20%5Bx-%5Cfrac%7B1%7D%7B7%7Dx%5E7%5D%5E1_0%3D)
![81 \pi [(1-\frac{1}{7}(1)^7)-(0-\frac{1}{7}(0)^7)]=](https://tex.z-dn.net/?f=81%20%5Cpi%20%5B%281-%5Cfrac%7B1%7D%7B7%7D%281%29%5E7%29-%280-%5Cfrac%7B1%7D%7B7%7D%280%29%5E7%29%5D%3D)
![81 \pi (1-\frac{1}{7}-0)=](https://tex.z-dn.net/?f=81%20%5Cpi%20%281-%5Cfrac%7B1%7D%7B7%7D-0%29%3D)
![81 \pi -\frac{81}{7}](https://tex.z-dn.net/?f=81%20%5Cpi%20-%5Cfrac%7B81%7D%7B7%7D)
the area is
![81 \pi-\frac{81}{7}](https://tex.z-dn.net/?f=81%20%5Cpi-%5Cfrac%7B81%7D%7B7%7D)
square units
It should be B. The dash-line triangle is half the size of the solid-lined triangle. You can find the scale factor by finding the ratio between the side lengths of the two triangles. The scale factor in this case is 2 because you are reducing thr solid-lined triangle by one half. If you are reducing by a fraction, it means that you are enlarging be the reciprical of the fraction. As reducing implies division and dividing a fraction is equivalent to multiplying by the reciprical. Thus the correct answer is B.
Answer:
12
Step-by-step explanation:
5^2 + x^2 = 13^2
x = 12
Answer:
Step-by-step explanation:On the given diagram you see two right triangles with equal hypotenuses and pair of equal angles.
Based only on the information given in the diagram you can conclude that triangles are congruent by HA theorem (the HA theorem states that "if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent").
Also you can conclude that triangles are congruent by AAS Postulate (the AAS Postulate states that "if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent").