Question

can someone help me with integrals and left hand sums?

Answer 1

What do you need? Do you have any specific questions?

Answer 2

it says estimate (x^2+1) d x with a left hand sum of n=3. the integral limits are 15 and 0.

Answer 3

Ok. We want to estimate the area under the curve using rectangles, so we need to know the height and width of each rectangle. "n = 3" means we want to use 3 rectangles, so
\[\Delta x =\frac{b-a}{n} = \frac{15-0}{3} = 5.\]
That is, the widths are each 5. The heights are given by the value of the function at the left endpoints. So we need f(0), f(5), and f(10).

I'm going to try drawing a picture!

Answer 4

but how do we calculate the heights?and the rectangles are invisible, right? cause for some of these problems, there are a graph. okay i'd appreciate a picture so much please!

Answer 5

We can always make a graph ourselves if we're not given one. Since the rectangles go up until the left-hand corner touches the function, the value of the function tells us the height. So the first rectangle has a height of f(0) = 0^2+1 = 1. The second one has a height of f(5) = 5^2+1 = 26. The last one is f(10) = 10^2+1 = 101.

Once we have that, add up the areas of each of them:
\[ \Delta x * f(0) + \Delta x * f(5) + \Delta x * f(10) = 5*1 + 5* 26 + 5* 101.\]

Answer 6

thank you sooooo much! but would it be the same if there was a right hand sum?what is the difference?

Answer 7

For a right hand sum, the rectangles go up until the right corner touches the function. Then the height is given by the value of the function there, so instead we'd use f(5), f(10), and f(15) for the heights.