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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Finding Area by the Limit Definition In Exercises 57-62, use the limit process to find the area of the region bounded by the graph of the function and the y-axis over the given y-interval. Sketch the region.

f ( y ) = y 2 , 0 y 5

To determine

To calculate: The area of the region bounded by the function f(y)=y2 and the interval is [0,5]

Explanation

Given: The provided function is,

f(y)=y2 and the interval is [0,5].

Formula used: The sum of squares of first n natural numbers is given by the formula as,

i=1ni2=n(n+1)(2n+1)6

Using upper endpoints area is,

A=limni=1nf(Mi)(Δx)

Here, Mi is the upper endpoints.

For a general quadratic equation f(x)=ax2+bx+c, coordinates of the vertex are,

(b2a,f(a))

Calculation: The graph of the region will be a parabola as the provided function is a quadratic. For the provided quadratic values of the coefficients are a=1,b=0 and c = 0. Now for findthe

coordinates of the vertex of parabola,

Consider the vertex of the provided quadratic equation (h,k)

Then,

h=b2a=02(1)=0

Now

k=f(a)=(0)2=0

So, the coordinates of the vertex are (0,0).

Now, as coefficient of x2 is positive so the parabola will open in the positive side of the x-axis.

Therefore, the required graph with the region is shown below,

Now, as the provided function f(y) is continuous and non-negative in the interval [0,5]. So, by the partition of interval into n subintervals each of equal width. Therefore,

Consider the width of each rectangle is (Δy) then,

The width of each rectangle is,

Δy=5(0)n=5n

The area can be calculated by the use of lower endpoints (mi) or upper endpoints (Mi). For this problem the upper endpoints are convenient.

Now, find upper endpoints (Mi).

Mi=0+5in=5in

So, the upper endpoint is Mi=5in

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