Question

Use the rectangles to approximate the area of the region. (Round your answer to three decimal places.)

The *x* *y*-coordinate plane is given. There is 1
curve, a shaded region, and 4 rectangles on the graph.

- The curve enters the window in the first quadrant, goes down
and right becoming less steep, passes through the point (1, 5),
passes through the point (5, 1), and exits the window almost
horizontally above the
*x*-axis. - The region below the curve, above the
*x*-axis, and between the values of 1 and 5 on the*x*-axis is shaded. - The segments of the
*x*-axis from*x*= 1 to 2, 2 to 3, 3 to 4, and 4 to 5 serve as the bases of the 4 rectangles. - The curve passes through the midpoint of the top of each of the 4 rectangles.

f(x) =

5 |

x |

[1, 5]

Give the exact area obtained using a definite integral.

Answer #1

we are given

interval is [1,5]

n=4

Firstly, we will find delta x

**Approximate Area:**

we can find mid point sum

now, we can plug values

**Exact Area:**

we can set up integral for area

**.............Answer**

Use the midpoint rule with 4 rectangles to approximate the area
of the region bounded above by y=sinx, below by the ?x-axis, on
the left by x=0, and on the right by ?=?

1- Find the area enclosed by the given curves.
Find the area of the region in the first quadrant bounded on the
left by the y-axis, below by the line above left
by y = x + 4, and above right by y = - x 2 + 10.
2- Find the area enclosed by the given curves.
Find the area of the "triangular" region in the first quadrant that
is bounded above by the curve , below by the curve y...

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?

1) Find the volume of the solid formed by rotating the region
enclosed by
y=e^(5x)+2, y=0, x=0, x=0.4
about the x-axis.
2) Use the Method of Midpoint Rectangles (do NOT use the
integral or antiderivative) to approximate the area under the curve
f(x)=x^2+3x+4 from x=5 to x=15. Use n=5 rectangles to find your
approximation.

Approximate the area under the graph of f(x) and above the
x-axis with rectangles, using the following methods with n=4.
f(x)=6x+4 from x=3 to x=5
a) use left endpoints
b)use right endpoints
c) average the answers in parts a and b
d) use midpoints

approximate the area under the graph of f(x) and above the
x-axis with rectangles, using the following methods with n=4.
f(x)=4x+5 from x=4 to x=6
A) use left endpoints
B) use right endpoints
C) average the answers in parts a and b
D) use midpoints

1. (a) Use the graph paper provided to sketch the region bounded
by the x-axis and the lines x + y = 4 and y = 3x.
(b) Shade the region you just drew above.
(c) Suppose that you were going to use Calculus to compute the
area of the shaded region in part (a) above. If you chose the
x-axis as your axis of integration, then how many integrals would
be needed to compute this area?
(d) Suppose that...

Approximate the area under the graph of f(x) and above the
x-axis with rectangles, using the following methods with n=4
f(x)=e^x+5 from x=-2 to x=2
(a) Use left endpoints.
(b) Use right endpoints.
(c) Average the answers in parts (a) and (b)
(d) Use midpoints.

Approximate the area under the graph of f(x) and above the
x-axis with rectangles, using the following methods with n=4.
f(x)=e^x+1 fromx=-2 to x=2
(a) Use left endpoints.
(b) Use the right endpoints.
(c) Average the answers in parts (a) and (b)
(d) Use midpoints.
The area, approximated using the left endpoints, is
The area, approximated using the right endpoints, is
The average of the answers in parts (a) and (b) is
The area, approximated using the midpoints, is

You are the foreman of the Bar-S cattle ranch in Colorado. A
neighboring ranch has calves for sale, and you are going to buy
some calves to add to the Bar-S herd. How much should a healthy
calf weigh? Let x be the age of the calf (in weeks), and let y be
the weight of the calf (in kilograms). x 1 5 11 16 26 36 y 39 47 73
100 150 200 Complete parts (a) through (e), given...

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