Wednesday
June 19, 2013

# Homework Help: Calculus

Posted by John on Thursday, March 7, 2013 at 12:03am.

Given that \displaystyle \int_0^4 x^3\sqrt{9+x^2} dx = a, what is the value of \lfloor a \rfloor?

• Calculus - Count Iblis, Thursday, March 7, 2013 at 11:55am

Put x = 3 sinh(t), then:

Integral from 0 to 4 of

x^3 sqrt{9+x^2} dx =

Integral from 0 to arcsinh(4) of

3^5 sinh^3(t) cosh^2(t) dt =

3^5 Integral from 0 to arcsinh(4) of

sinh(t) [cosh^2(t) - 1] cosh^2(t) dt =

3^5 Integral from 0 to arcsinh(4) of

[cosh^2(t) - 1] cosh^2(t) dcosh(t)

Putting cosh(t) = u gives:

3^5 Integral from 1 to sqrt(17) of

[u^2 - 1] u^2 du =

3^5 {sqrt(17)[1/5 17^2 - 1/3 17] + 2/15 }

You then need to round this off to below.

Related Questions

Calculus - Given that \displaystyle \int_0^4 x^3\sqrt{9+x^2} dx = a, what is the...
Algebra - What is \left \lfloor ( 3 + \sqrt{5} ) ^3 \right \rfloor?
calculus - It is known that if m &lt;= f(x) &lt;= M for a &lt;= x &...
ap calculus - Which of the following definite integrals gives the length of y = ...
calculus - prove that d/dx 4x .&#8730;(x + &#8730;x) = 6x+5 (x)1/2/&...
Calculus - Use the Midpoint Rule with the given value of n to approximate the ...
Calculus - Use the Midpoint Rule with the given value of n to approximate the ...
math - (a) Use the Midpoint Rule and the given data to estimate the value of the...
Calculus - Find the exact coordinates of the centroid. y = sqrt[x], y = 0, x = 9...
calc check: average value - Find the average value of the function "f(x) = ...