Let f : [a, b] → R be a differentiable function and let P be a partition of [a, b]. Show that L(P, f′) ≤ f(b) − f(a). Hint: Use the Mean-Value Theorem. (Recall that, if P...

Let f : [a, b] → R be a differentiable function and let P be a partition of [a, b]. Show that L(P, f′) ≤ f(b) − f(a). Hint: Use the Mean-Value Theorem. (Recall that, if P...

Integrate with whole solution

Does the integral converge or diverge? ∫1∞(lnx)2/x1.2 dx

Let R be the region in R^2 enclosed by the curve y = x^2 + 3 and the line y = 4x + 3. a) Write the area of R as a definite integral. b) Calculate...

suppose that f(x) is an even function and let ∫0^1 f(x)dx = 5 and ∫0^7 f(x)dx = 1. What is ∫-7^-1 f(x)dx? This is my work I expand ∫-7^-1 f(x)dx to F(-1) - F(-7) F(-1)...

Using the techniques of trig substitution : ∫×√(4+x2 ) dx and ∫×√4-x2 ) dx How can I solve these using trig substitution?

The question asks to estimate the area under the graph given the following table: | x | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 | | y | 4 | 10 | 8 | 6 | 14 | 10 | 12 | a) Use Simpson's...

Area bounded by y2=x-1 and the line y=x-3

A ball drops from a height of 20 feet. Each time it hits the ground, it bounces up 40 percents of the height it fall. Assume it goes on forever, find the total distance it travels.

To find the volume of a right circular cone with base radius r and height h, the cone is divided into n frustums of equal heights. The volume of each frustum is approximated as if it were a circular...

Zero to infinity∑ (-1)n * ((5n2+7)/(7n2+10))^n I think I have to use the root test but not sure. What test do I use? If it is the root test, does the square root cancel...

Find the area bounded by the curve: y = x(x-1)^2, the line y = 2, and the y - axis

How do I do this ?

I dont understand this because when I find the intersection points which is x=0 and x=1 , how do I know which is the outer and lower function ?

Basically what the questions states. Thanks

I'm having a lot of trouble with the set up and integration. Please help me with this problem, thank you. I know you are supposed to use the formula integral(2pi * y * sqrt(1+(y')^2)).

∫ (1)/(sqrt(52x -1))dx Please show me steps. Im having a hard time figuring this out

I still havent learned anthing about logarithm derivatives yet so i have no idea how to proof this even i know the answer is ln2. So far with the definition of integral i transformed...

∫ba (2+x-x2)dx

2∫ab f(x)f'(x)dx=[f(b)]2-[f(a)]2

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