What is the primary purpose of dividing the area under a curve into rectangles?

What is the formula for calculating the width of each rectangle in a partition?

Suppose $\sum _{k=1}^{12}{x}_k=18$ and $\sum _{k=1}^{12}{y}_k=-7$. Compute the value of $\sum _{k=1}^{12}(x_k-y_k+3)$.

Evaluate the summation Σ from k=1 to 4 of 3k using the constant multiple rule.

Calculate the left and right Riemann sums for ${\int }_1^4(2-x)dx$ with $n=6$ subintervals.

What is a definite integral?

Find the exact value of the following definite integral:

${\int }_0^{\frac{\pi }{2}}{\cos }^{4}xdx$

Evaluate the integral.

${\int }_2^8\frac{x^{\frac{7}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}dx$

Find the derivative of the integral \( \int_{0}^{x} (3t^2 + 2t) \, dt \) with respect to x.

A car accelerates from rest at a rate of \( a(t) = 3t \) m/s². Using the Fundamental Theorem of Calculus, find the velocity of the car at \( t = 4 \) seconds.

Let $g\left(x\right)={\int }_2^x{t}^2(t-1)(t-5)dt$. Find the $x$-coordinates of the critical points of $g\left(x\right)$ and determine the intervals where $g$ is increasing or decreasing.

Find the average value of the function $f\left(x\right)=\sin x$ on the interval $[0,\pi ]$.

Determine the average value of f(x) = x^3 on the interval [1, 2].

Find the average value of the function $g\left(\theta \right)=\frac{1}{1-\cos \theta }$ on the interval $[\frac{\pi }{6},\frac{\pi }{3}]$.