Question

calculus 2 - explanation

Answer 1

Calculus 2 is the second portion of calculus and mainly focuses on integrals, series and sequences. Is that explained well?

Answer 2

Sorry, I meant from the specific pic Im' posting

Answer 3

That pic must be a large one, since it didn't load for me. Could you possibly explain in your own words just what it is that you want from your helpers in this particular case?

Answer 4

All i see is a definition.

Answer 5

Sorry, I was asking for someone to explain that piece to me. I was asking for clarification since I didn't fully grasp it.

Answer 6

The Draw button is not working for me for a couple of days and so I will attempt to explain without a drawing.

When we integrate a function f(x) between the limits a and b, we are computing the area under the curve f(x) and the x axis bounded by the vertical line y = a on the left and y = b on the right. When we evaluate an integral between constant limits such as a and b it is called a definite integral. It is a number as opposed to integrating f(x) which yields a function.

Now the Riemann sum:

Answer 7

We can split the interval [a, b] into n equal parts and draw a vertical line at each point until it meets the curve f(x). Now you will have n equal-width rectangles. The width of the rectangle will be (b-a)/n. The height of the rectangle can be computed in two ways. We can take the height of the rectangle to be where the vertical line from the left end-point of the sub-interval meets the curve or we can take the height of the rectangle to be where the vertical line from the right end-point of the sub-interval meets the curve.

For left-hand Riemann sum:
The area of each rectangle is: (b-a)/n * f(left end-point)
Area of all rectangles = summation of all small rectangles.
And Riemann sum is the limit as n goes to infinity. That is, we divide [a,b] into infinite rectangles so we get the exact area under the curve. This area is the same as the one given by the definite integral discussed earlier.

You do the Riemann sum with the right end-point of each sub-interval you get the Right-Hand sums.

Answer 8

@TachiHere : Ranga is on track in explaining what a Riemann sum is and what it's used for. In evaluating Riemann sums, we generally have a choice of 3 different approaches: right-hand sums, left-hand sums and midpoints. The given definition is needlessly complicated.

Answer 9

The point of using Riemann sums to evaluate the area under a curve between x=a and x=b is that they return results equivalent to definite integrals (which are often a lot faster than Riemann sums to evaluate). In fact, discussions of Riemann sums lead up to definitions of the definite integral.

Answer 10

In my first reply, the vertical lines should read x = a and x = b.

Answer 11

thanks a lot ranga, i think i kinda get it now