Question

Use the Riemann Sum (limit process) to find the area between y = x 2 + 2x and x-axis over the interval [0, 2].

Answer #1

2. Using the limit definition of the integral (Riemann Sums),
Find the area under the curve from [1, 11] y = 2x 2 + 4x + 6 Recall
that Pn i=1 i = n(n+1) 2 and Pn i=1 i 2 = n(n+1)(2n+1) 6

Let f(x) = e^x. Evaluate a right Riemann sum for the interval
[−1, 1] for n = 4. You should include a picture of the
corresponding rectangles and state if this is an under or over
approximation of the area beneath the graph of f, above the x-axis
and between x = −1 and x = 1. In your solution, you should write
out all terms that go into the Riemann sum.

(a) Find the Riemann sum for
f(x) = 3
sin(x), 0 ≤ x ≤
3π/2,
with six terms, taking the sample points to be right endpoints.
(Round your answers to six decimal places.)
R6 =
(b) Repeat part (a) with midpoints as the sample points.
M6 =
Express the limit as a definite integral on the given
interval.
lim n → ∞
n
7xi* +
(xi*)2
Δx, [3, 8]
i = 1
8
dx
3

1. Find the area between the curve f(x)=sin^3(x)cos^2(x) and y=0
from 0 ≤ x ≤ π
2. Find the surface area of the function f(x)=x^3/6 + 1/2x from
1≤ x ≤ 2 when rotated about the x-axis.

1. Evaluate the Riemann sum for
f(x) = 2x − 1, −6 ≤ x ≤ 4,
with five subintervals, taking the sample points to be right
endpoints.
2. sketch a graph
3. Explain.
The Riemann sum represents the net area of the rectangles with
respect to the .....

USING ITERATED INTEGRALS, find the area bounded by the
circle x^2 + y^2 = 25,
a.) the x-axis and the parabola x^2 − 2x = y
b.) y-axis and the parabola y = 6x − x^2
b.) (first quadrant area) the y-axis and the parabola x^2 − 2x =
y

find the area between the curves, y=1-x^2 and y= x^3 + 2x^2
+1

Express the given integral as the limit of a Riemann sum but do
not evaluate: the integral from 0 to 3 of the quantity x cubed
minus 6 times x, dx.

Estimate the area of the region bounded between the curve f(x) =
1 x+1 and the horizontal axis over the interval [1, 5] using a
right Riemann sum. Use n = 4 rectangles first, then repeat using n
= 8 rectangles. The exact area under the curve over [1, 5] is ln(3)
≈ 1.0986. Which of your estimates is closer to the true value?

If f(x) = 2x2 − 7, 0 ≤ x ≤ 3, find the Riemann sum with n = 6,
taking the sample points to be midpoints. What does the Riemann sum
represent? Illustrate with a diagram.

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