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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Finding Area by the Limit Definition In Exercises 47-56, use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region.

y = 27 x 3 , [ 1 , 3 ]

To determine

To calculate: Area of the region bounded by y=27x3, in the interval [1,3] and the x-axis.

Explanation

Given: y=27x3, in the interval [1,3].

Formula used: Formula for the sum of cubes of first n natural numbers:i=1ni3=n2(n+1)24

Formula for the sum of squares of first n natural:i=1ni2=n(n+1)(2n+1)6

The sum of a constant n times is written as:i=1nc=nc

Formula for the sum of first n natural numbers:i=1ni=n(n+1)2

Using right endpoints area is written as:Area=limni=1ny(Mi)(Δx), where, Mi are the right endpoints.

Calculation: Function y is continuous and non-negative in the interval [1,3].

Partition the interval into n subintervals each of width Δx:

Δx=3(1)n=2n

Area can be calculated by left endpoints (mi) or right endpoints (Mi).

Right endpoints (Mi) are:

1+2in=2i+nn,

i=1,2,3,.......,n

Area=limni=1ny(Mi)(Δx) where, Mi are the right endpoints, and Mi=2i+nn.

Area=limni=1ny(2i+nn)(Δx)

Use value of y(2i+nn) and Δx.

Area=limni=1n(27(2i+nn)3)(2n)

Split the expression in parts to use summation formulas:

Area=limni=1n54nlimni=1n2n4(2i+n)3

Expand the second sum:

Area=limni=1n54nlimni=1n2n4(8i3+n3+12i2n+6in2)

Factor out 2n4 from the second sum and expand the second sum to use summation formulas:

Ar

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