how do I integrate x^4 using the integral definition? By definition I mean the definition of an integral, as in a limit of a Riemann sum. I'm teaching myself about integrals and I'm not sure how to solve this problem without running into issues. I know using the power rule that the answer is supposed to be 618.6, but im not sure how to do it the long way.
Here is my work so far:
"integrate $∫_2^5 x^4$ using the definition:"
$$∫_a^b f(x)\,dx)=∑_{i=1}^n f(x_i)Δx $$
$$Δx=(b-a)/n,\enspace
x_i=a+iΔx $$
a=2
b=5
Δx=3⁄2
xi=2+3i⁄n
f(xi)=(2+3i⁄n)4
f(xi)=(16+(96i)⁄n+(216i2)⁄n2>+(216i3)⁄n3+(81i4)⁄n4
∑ni=1[3(16+96i+216i2+216i3+81i4)/n5]
∑ni=1[48/n+288i/n2+648i2/n3+648i3/n4+243i4/n5
48/n∑ni=1 + 288/n^2∑ni=1 i + 648/n^3 ∑ni=1 i^2 + 648/n^4∑ni=1 i^3 + 243/n^5∑ni=1 i^4
48+(288/n^2)((n(n+1)/2)+(648/n^3)(n(n+1)(2n+1)/6)+(648/n^4)((n(n+1/2))^2+(243/n^5)(n(n+1)(2n+1)/6)^2
 A: By the definition we will have $\Delta x=\frac{3}{n}$ and $x^i=2+\frac{3}{n} i.$ So now we will deal with the expression:
$$\lim\limits_{n \to \infty}\sum_{i=1}^n(2+\frac{3}{n} i)^4(\frac{3}{n})$$ After you have expanded the inside of the summation, dealt with the tedious algebra to simplify and applied the summation formulas for $i, i^2, i^3$ and $i^4$ you will have the following:
$$\lim\limits_{n \to \infty}\frac{3}{n}(16n+\frac{96}{n}(\frac{n(n+1)}{2})+\frac{216}{n^2}(\frac{n(n+1)(2n+1)}{6})+\frac{216}{n^3}(\frac{1}{4}n^2(n+1)^2)+\frac{81}{n^4}(\frac{1}{30}n(n+1)(2n+1)(3n^2+3n-1))$$ Which simplifies to:
$$\lim\limits_{n \to \infty}\frac{3}{10n^4}(2062n^4+3045n^3+1170n^2-27)$$ Which at last becomes:
$$\lim\limits_{n \to \infty}618.6+\frac{913.5}{n}+\frac{351}{n^2}-\frac{8.1}{n^4}=618.6$$ Which is the correct answer since as $n \to \infty $, the last $3$ terms all goes to $0$ and we're only left with the constant. Setting up the Riemann Integral is pretty easy if you understand the definition, the only tricky thing is the actual computation itself, it can get ridiculous when you're dealing with harder functions, this is obviously why we don't really use the definition when integrating a function but instead we have wonderful rules which allows us to skip all the horrible algebra/arithmetic, similarly to when we differentiate functions we don't use the formal definition of the derivative every time, we use power rule, chain rule etc.
