To take a right-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 will have size 2. On the first sub-interval ([0,2]), we do the following: - Go to the
**right endpoint** of the sub-interval (2). - Go straight up until you hit the function.
Then 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, and calculate the area of the rectangle:
The area of this rectangle is (height) ⋅ (width) = 5 ⋅ 2 = 10 Now we do the same stuff on the second sub-interval ([2,4]). - Go to the
**right endpoint** of the sub-interval (4). - Go straight up until you hit the function.
Then figure out the *y*-value of the function where you hit it (*f* (4) = (4)^{2} + 1 = 17). - Make a rectangle whose base is the subinterval and whose height is the
*y*-value you just found, and calculate the area of the rectangle:
The area of this rectangle is (height) ⋅ (width)17 ⋅ 2 = 34 Adding the areas of the rectangles together, we see that the rectangles cover an area of size 10 + 34 = 44: This is an overestimate for the actual area of *R*, since the rectangles cover *R* and then some. We can get a better estimate by dividing the interval [0,4] into more sub-intervals and using more rectangles. |