To take a left-hand sum we first divide the interval in question into sub-intervals of equal size. Since we're looking at the interval [0,4], each sub-interval has size 2. On the first sub-interval, [0,2], we do the following: - Go to the
**left endpoint** of the sub-interval (0). - Go straight up until you hit the function.
Figure out the *y*-value of the function where you hit it (*f* ( 0 ) = (0)^{2} + 1 = 1). - Make a rectangle whose base is the subinterval and whose height is the
*y*-value you just found:
Finally, calculate the area of this rectangle: (height) ⋅ (width) = 1 ⋅ 2 = 2 Now we do the same stuff on the second sub-interval, [2,4]. - Go to the
**left endpoint** of the sub-interval (2).
- Go straight up until you hit the function.
Figure out the *y*-value of the function where you hit it (*f** *( 2 ) = (2)^{2} + 1 = 5). - Make a rectangle whose base is the sub-interval and whose height is the
*y*-value you just found:
Finally, calculate the area of this rectangle: (height) ⋅ (width) = 5 ⋅ 2 = 10 Adding the areas of the rectangles together, we see that the rectangles cover an area of size 12: This is less than the area we're trying to find, because the two rectangles don't cover all of *R*. That's ok, because we're only estimating. In order to find a better estimate, we can use more rectangles. We start by dividing the original interval into more sub-intervals (each sub-interval will now be smaller). On each sub-interval we do the same stuff we did before. |