Question

definite integral with u sub

Answer 1

\[\int\limits_{0}^{1}\sqrt{x}(1+x)dx\]

Answer 2

u know indefinte integration

Answer 3

why the need of a u sub?

Answer 4

looks like its already done then

Answer 5

well you already have the answer from wolfram

Answer 6

so what is the question you have about the process?

Answer 7

well yea, i just dont get their u sub method, where the first u^2 comes form

Answer 8

its just a working of the substitution ....

u = sqrt(x)

du = 1/2sqrt(x) dx

2 sqrt(x) du = dx

but its already defined that u = sqrt(x)

2u du = dx

Answer 9

u^2 = (sqrt(x))^2 as well

Answer 10

therefore:

2u u (u^2 + 1) du is the sub

Answer 11

was there a reason for a u sub to start with?

Answer 12

i guess not i distributed the sqrtx through then since its addition you can integrate seperatly

Answer 13

maybe you would like this problem, a metal plate of dendity δ(y)=1+y,has a shape bounded by the curve y = √x, the x-axis, and the line x = 1. find the mass

Answer 14

a shape bounded by the curve y=sqrtx

Answer 15

m=density*volume. \[\int\limits_{0}^{1}(1+y)(volume)dy\]

Answer 16

on the solution sheet for my exam review it states volume as 1-y^2, while if you solve for y in y=sqrtx you just have y^2

Answer 17

we have a metal plate in a shape, hose density at any point is defined as d(y)

mass is equal to the area*density .... in 3d objects i spose volume is proper. in some classes they simply define area as 2d volume

Answer 18

a multiple integration of dy dx is what im considering

Answer 19

i just dont understand where solving for x in the bounds function a 1-y^2 came up, maybe through polhatogean theorem

Answer 20

Pythagorean*

Answer 21

x = 0 to 1
y = 0 to sqrt(x)

and for any point on the plate, we have a density value of (1+y)

\[\int_{0}^{1}\int_{0}^{\sqrt{x}}(1+y)dy~dx\]
integrate the inside
\[\int_{0}^{1}\left(\int_{0}^{\sqrt{x}}(1+y)dy\right)~dx\]
\[\int_{0}^{1} (\sqrt{x}+\frac12x)~dx\]
then wrap up whats left over

Answer 22

5/6 is the answer, but thats not the correct way to do it in terms of the question.

Answer 23

they did dx dy