Video: Pack 2 • Paper 2 • Question 20

Pack 2 • Paper 2 • Question 20

03:16

Video Transcript

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 squared.

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.

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