The graph of 𝑦 equals negative 𝑥
squared plus eight 𝑥 minus five is shown. Part a) Calculate an estimate for
the area under the curve between the lines 𝑥 equals two and 𝑥 equals six. Use four strips of equal width. Part b) Is the answer to part a an
overestimate or an underestimate?
Our first step is to split the
graph into four strips of equal width between 𝑥 equals two and 𝑥 equals six. We now have five 𝑥-coordinates and
we need to find the corresponding 𝑦-values from the graph. When 𝑥 is equal to two, 𝑦 is
equal to seven. When 𝑥 is equal to three, 𝑦 is
equal to 10. When 𝑥 is equal to four, 𝑦 is
equal to 11. When 𝑥 is equal to five, 𝑦 is
equal to 10. And finally, when 𝑥 is equal to
six, 𝑦 is equal to seven.
We could also have found these
values of 𝑦 by substituting our 𝑥-coordinates into the equation 𝑦 equals negative
𝑥 squared plus eight 𝑥 minus five. We have now created four
trapeziums, which we can use to calculate an estimate for the area under the curve
between 𝑥 equals two and 𝑥 equals six. We can calculate the area of any
trapezium using the formula a half multiplied by 𝑎 plus 𝑏 multiplied by ℎ, where
𝑎 and 𝑏 are the parallel sides of the trapezium and the ℎ or height is the
distance between the parallel sides.
Trapeziums A and D have parallel
sides of lengths seven and 10. Therefore, we can calculate the
area of these trapeziums by multiplying a half by seven plus 10 by one. Seven plus 10 is equal to 17 and a
half of 17 is 8.5. 8.5 multiplied by one is 8.5. Trapeziums B and C have parallel
sides of lengths 10 and 11. Therefore, the area of these
trapeziums is a half multiplied by 10 plus 11 multiplied by one. This is equal to 10.5. Therefore, an estimate for the area
under the curve can be calculated by adding 8.5, 8.5, 10.5, and 10.5. This is equal to 38 units
The second part of our question
asked us to decide whether the first answer was an overestimate or an
underestimate. Well, in this case, our answer is
an underestimate as the four trapeziums A, B, C, and D were all below the curve. The area under the curve would be
slightly larger than 38.