We cut the original interval into 4 sub-intervals of length 2. On each sub-interval, the height of the rectangle is given by the value of the function at the right-most endpoint of that sub-interval. On the sub-interval [0,2] the height of the rectangle is *f** *(2) = 6. On the sub-interval [2,4] the height of the rectangle is *f* (4) = 18. On the sub-interval [4,6] the height of the rectangle is *f* (6) = 38. On the sub-interval [6,8] the height of the rectangle is *f* (8) = 66. We could find the area covered by all 4 rectangles by finding the area of each rectangle (multiplying each height by 2) and adding them: *f* (2)(2) + *f* (4)(2) +* **f* (6)(2) +* **f* (18)(2)
The distributive property says that we'll get the same answer if we first add all the heights and then multiply their sum by 2: [*f** *( 2 ) + *f *(4) +*f*(6) +*f*(8)](2). We think this second option is a lot easier. You only need to hit the multiplication button once! The final answer is [6 + 18 + 38 + 66](2) = 256. To summarize: to quickly find a RHS, take the value of the function at the right endpoint of each sub-interval and find the sum of these values. Then multiply the sum by the width of a sub-interval/rectangle. The value of the function at the left-most endpoint of the original interval will never be used. |