integration Q's (1 Viewer)

[-pari-]

integrating things like...

1) x^3(x^4 -1)^8

2) x/ (3x^2 - 4)^2

no idea how to do it.

also, finding the volume of an area between two curves thats been rotated

3) Find the area between y = x^2, y = x^3. find the volume of the solid formed if this area is rotated about the x axis.

answer: 2(pi)/5

4) the area bounded by the two curves y = x^2 and y^2 = x is rotated about the xaxis. find the volume of the solid that is generated.
answer: 3(pi)/10

finding the area where a semi-circle is involved:

5) find the exact area enclosed between the curve y = rt(4 - x^2) and the line x - y + 2 = 0.

integrating like usual, i get 10. the answer however is (pi -2)


any help would be great.

 

[jemsta]

for the first question, i think the only way you can do it is by substitution, which i dont think you need to know if youre doing only mathematics 2unit.
is the answer by any chance (x^4-1)^9/36+C?

 

[bos1234]

Ya Thats What I Got Too

 

[SoulSearcher]

The first two questions you would have to use the substitutions u =x⁴ and u = x² to find the integral.

3 and 4 I'm working on atm, but I think you would first find the intersection points, then integrate both curves seperately with the terminals being the x-coordinates of the intersection points, then subtract one volume form the other, but I'm not too sure if that actually works out, find someone to try and solve it.

5, draw the graph, which shows that the intersection points lie on the x and y axes, hence you are able to find the area by calculating the area of half a semicircle and the area of the triangle made by the line x - y + 2 = 0 and the x and y axes, then subtract the area of the triangle from the semicircle and you get your answer.