Since we're looking at the interval [0, 4], each sub-interval has length 1. The first sub-interval is [0,1]: - Go to the
**left endpoint** of the sub-interval (0).
- Go straight up until you hit the function and 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:
Calculate the area of the rectangle: (height) ⋅ (width) = 1 ⋅ 1 = 1 The second sub-interval is [1,2]: - Go to the
**left endpoint** of the sub-interval (1).
- Go straight up until you hit the function and figure out the
*y*-value of the function where you hit it (*f* (1) = (1)^{2} + 1 = 2).
- Make a rectangle whose base is the subinterval and whose height is the
*y*-value you just found:
Calculate the area of the rectangle: (height) ⋅ (width) = 2 ⋅ 1 = 2 The third sub-interval is [2, 3]: - Go to the
**left endpoint** of the sub-interval (2).
- Go straight up until you hit the function and 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 subinterval and whose height is the
*y*-value you just found:
Calculate the area of the rectangle: (height) ⋅ (width) = 5 ⋅ 1 = 5 The last sub-interval is [3, 4]: - Go to the
**left endpoint** of the sub-interval (3).
- Go straight up until you hit the function and figure out the
*y*-value of the function where you hit it (*f* (3) = (3)^{2} + 1 = 10).
- Make a rectangle whose base is the subinterval and whose height is the
*y*-value you just found:
Calculate the area of the rectangle: (height) ⋅ (width) = 10 ⋅ 1 = 10 The area covered by these four rectangles is 1 + 2 + 5 + 10 = 18. We're still not quite covering the whole area we want, but we're closer. |