Which of the following equals Beta function B(m, n)?
Standard identity relating Beta and Gamma functions: B(m, n) = $\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$.
Evaluate $\int_0^{\infty} x^3 e^{-x} dx$.
By definition of Gamma function, $\int_0^{\infty} x^{n-1} e^{-x} dx = \Gamma(n)$. Here n-1 = 3 => n = 4. $\Gamma(4) = 3$! = 6.
Which of the following definite integrals have a value equal to 0?
Integrals 1, 2 (odd functions over symmetric intervals) and 3 ($\sin(\pi)-\sin(0)=0) all evaluate$ to 0.
Calculate the volume of a sphere of radius R = 3.
Volume V = (4/3) $\pi R^3 = (4/3) \pi (27) = 36 \pi \approx 36 * 3.14159 = 113.097 \approx 113.1$.
Evaluating $\int_0^1 \int_0^1 \int_0^1 (x + y + z) dx dy dz yields$:
Integral = $\int_0^1 dx \int_0^1 dy \int_0^1 dz x +$... = 1/2 + 1/2 + 1/2 = 3/2 = 1.5.
