To determine
a)
To evaluate:
The Riemann sum for fx
Answer
1.25
Riemann sum using right endpoints for the sample points is an over estimate of the area under the graph of fx=x2-x
Explanation
Concept:
Definition of Riemann sum:
∑i=1nfxi* ∆x
is called as Riemann sum
Where, ∆x= b-an, xi=a+i ∆x, &xi*∈xi-1, xi
Riemann sum is the sum, of the areas of the rectangles that lie above the x-axis and the negative of the areas of the rectangles that lie below the x-axis
Given:
fx=x2-x, 0≤x≤2
Calculation:
First, write the expression of the Riemann sum
Four subintervals are given, that means, n=4
Taking the sample points as the right endpoints means that
The right endpoints are 0.5, 1, 1.5, 2
By definition of Riemann sum
∑i=1nfxi ∆x=∑i=14fxi ∆x
So now calculate ∆x,
∆x= b-an=2-04=12
Substitute this value in Riemann sum expression,
=∑i=14fxi12
=∑i=14xi2-xi12
=12f0.5+f1+f1.5+f2
=120.52-0.5+12-1+1.52-1.5+22-2
=12-0.25+0+0.75+2
=122.5
=1.25
Draw the diagram of Riemann sum for right endpoints
Riemann sum using right endpoints for the sample points is an over estimate of the area under the graph of fx=x2-x
Conclusion:
Therefore,
∑i=1nfxi ∆x≈1.25
Riemann sum using right endpoints for the sample points is an over estimate of the area under the graph of fx=x2-x
To determine
b)
To calculate:
The value of the integral ∫02x2-xdx
Answer
23
Explanation
Concept:
Definition of Definite integral:
∫abfxdx=limn→∞∑i=1nfxi ∆x
Where,
∆x= b-an and xi=a+i ∆x
Formula:
∑i=1ni2=nn+12n+16
∑i=1ni=nn+12
Calculation:
Let
∫abx2-xdx=limn→∞∑i=1nxi2-xi ∆x
Find out ∆x &xi,
∆x= b-an
= 2-0n
∆x= 2n
xi=a+i ∆x
=0+i·2n
xi=2in
Substitute the values of ∆x &xi in evaluating the limit
∫02x2-xdx=limn→∞∑i=1n2in2-2in2n
=limn→∞2n∑i=1n4i2n2-2in
=limn→∞2n∑i=1n4i2n2-∑i=1n2in
=limn→∞2n4n2∑i=1ni2-2n∑i=1ni
Substituting for the sums,
=limn→∞2n4n2·nn+12n+16-2n·nn+12
=2limn→∞2n2·n+12n+13-1n·n+1
=2limn→∞23·n+12n+1n·n-1n·n+1
=2limn→∞23·n+1n·2n+1n-n+1n
=2limn→∞23·1+1n·2+1n-1+1n
Simplify the limit,
=223·1+0·2+0-1+0
=243-1
=23
Conclusion:
Therefore,
∫02x2-xdx=23
To determine
c)
To check:
∫02x2-xdx=23
Answer
23
Explanation
Concept:
The Fundamental Theorem of Calculus-Suppose f is continuous on [a, b] then
∫abfxdx=Fb-F(a),F is an Antiderivative of f i.e F'=f
Calculation:
Consider, fx=x2-x
Take antiderivative, and by using power rule of integration,
Fx=x33-x22
By Fundamental Theorem of Calculus,
∫02x2-xdx=F2-F0
=233-222-033+022
=83-42
=23
Conclusion:
Therefore, By Fundamental Theorem of Calculus,
∫02x2-xdx=23
To determine
d)
To draw:
Graph of the function fx=x2-x
Answer
Explanation
Concept:
The net area under the curve is the area under the curve under the x-axis subtracted from the area under the curve above x-axis
Calculation:
Graph of the function fx=x2-x is
The net area under the curve is
∫02x2-xdx=A2-A1
Conclusion:
Therefore, by graph of the function fx=x2-x
The net area under the curve is
∫02x2-xdx=A2-A1