Use geometry and properties of integrals to evaluate the following definite inte

Tabansi 2021-10-14 Answered
Use geometry and properties of integrals to evaluate the following definite integrals.
048xx2dx
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Expert Answer

Asma Vang
Answered 2021-10-15 Author has 93 answers
Complete the square: (8xx2)=(x28x+1616)
(8xx2)=(x4)2+16
Substitute this integral, we get:
048xx2dx=04(x4)2+16dx
Apply u-substitution
u=x-4
du=dx
for x=0,u=-4;x=4,u=0
Thus the integral in terms of u is given by:
048xx2dx=40u2+16du
Use trigonometric substitution
u=4sin(v)
du=4cos(v)
for u=4,v=π2;u=0,v=0
The integral in terms of v is:
π2016sin2v+16(4cosv)dv
4π2016(1sin2v)cosvdv
4π2016cos2vcosvdv
4π204cosvcosvdv
16π20cos2vdv
16π201+cos(2v)2dv
8[v+12sin(2v)]π20
8[(0+0)(π2+12sin(π))]
4π
Answer: 4π
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