정적분과 급수(구분구적법)

For each of the three graphs:

a. Obtain a lower bound $L(A)$ for the area enclosed by the curve by adding the areas of the squares enclosed completely by the curve. b. Obtain an upper bound $U(A)$ for the area by adding to $L(A)$ the areas $B(A)$ of the squares enclosed partially by the curve.

In the previous exercise, explain why $L(A)$ gets no smaller while $U(A)$ gets no larger as the squares are subdivided into four boxes of equal area.

A fundamental calculus technique is to use _____ to refine approximations to get an exact answer.

What is the upper bound in the summation

$\displaystyle \sum_{i=7}^{14} (48i-201)$?

This section approximates definite integrals using what geometric shape?

T/F: A sum using the Right Hand Rule is an example of a Riemann Sum.

In Exercises 5–12, write out each term of the summation and compute the sum.

$\displaystyle \sum_{i=2}^{4} i^2$

In Exercises 5–12, write out each term of the summation and compute the sum.

$\displaystyle \sum_{i=-1}^{3} (4i-2)$

In Exercises 5–12, write out each term of the summation and compute the sum.

$\displaystyle \sum_{i=-2}^{2} \sin (\pi i/2)$

In Exercises 5–12, write out each term of the summation and compute the sum.

$\displaystyle \sum_{i=1}^{10} 5$

In Exercises 5–12, write out each term of the summation and compute the sum.

$\displaystyle \sum_{i=1}^{5} \frac1i$

In Exercises 5–12, write out each term of the summation and compute the sum.

$\displaystyle \sum_{i=1}^{6} (-1)^ii$

In Exercises 5–12, write out each term of the summation and compute the sum.

$\displaystyle \sum_{i=1}^{4} \left(\frac{1}{i} - \frac{1}{i+1}\right)$

In Exercises 5–12, write out each term of the summation and compute the sum.

$\displaystyle \sum_{i=0}^{5} (-1)^i\cos (\pi i)$

In Exercises 13–16, write each sum in summation notation.

$3+6+9+12+15$

In Exercises 13–16, write each sum in summation notation.

$-1+0+3+8+15+24+35+48+63$

In Exercises 13–16, write each sum in summation notation.

$\displaystyle \frac12+\frac23+\frac34+\frac45$

In Exercises 13–16, write each sum in summation notation.

$\displaystyle 1-e+e^2-e^3+e^4$

In Exercises 29–34, a definite integral

$\displaystyle \int_a^b f(x) \ dx$ is given.

(a) Graph $f(x)$ on $[a,b]$. (b) Add to the sketch rectangles using the provided rule. (c) Approximate $\displaystyle \int_a^b f(x) \ dx$ by summing the areas of the rectangles.

$\displaystyle \int_{-3}^3 x^2\ dx$, with 6 rectangles using the Left Hand Rule.

In Exercises 29–34, a definite integral

$\displaystyle \int_a^b f(x) \ dx$ is given.

(a) Graph $f(x)$ on $[a,b]$. (b) Add to the sketch rectangles using the provided rule. (c) Approximate $\displaystyle \int_a^b f(x) \ dx$ by summing the areas of the rectangles.

$\displaystyle \int_{0}^2 (5-x^2)\ dx$, with 4 rectangles using the Midpoint Rule.