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The diagram shows the curve with equation $y = 3 + 2x - x^2$ - Scottish Highers Maths - Question 1 - 2018

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Question 1

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The diagram shows the curve with equation $y = 3 + 2x - x^2$. Calculate the shaded area.

Worked Solution & Example Answer:The diagram shows the curve with equation $y = 3 + 2x - x^2$ - Scottish Highers Maths - Question 1 - 2018

Step 1

State an integral to represent the shaded area

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Answer

To calculate the shaded area under the curve, we need to set up the integral over the interval determined by the x-intercepts. Thus, the integral can be expressed as:

∫−13(3+2x−x2) dx\int_{-1}^{3} (3 + 2x - x^2) \, dx

Step 2

Integrate

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Answer

The integral of the function is:

∫(3+2x−x2) dx=3x+x2−x33+C\int (3 + 2x - x^2) \, dx = 3x + x^2 - \frac{x^3}{3} + C

Step 3

Substitute limits

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Answer

Next, we substitute the limits from −1-1 to 33 into the integrated function:

[3(3)+(3)2−(3)33]−[3(−1)+(−1)2−(−1)33]\left[ 3(3) + (3)^2 - \frac{(3)^3}{3} \right] - \left[ 3(-1) + (-1)^2 - \frac{(-1)^3}{3} \right]

Step 4

Evaluate integral

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Answer

Now we calculate:

For the upper limit (x=3x = 3):

3(3)+9−9=93(3) + 9 - 9 = 9

For the lower limit (x=−1x = -1):

3(−1)+1+13=−3+1+13=−2+13=−63+13=−533(-1) + 1 + \frac{1}{3} = -3 + 1 + \frac{1}{3} = -2 + \frac{1}{3} = -\frac{6}{3} + \frac{1}{3} = -\frac{5}{3}

Thus, the area is:

9−(−53)=9+53=273+53=3239 - \left(-\frac{5}{3}\right) = 9 + \frac{5}{3} = \frac{27}{3} + \frac{5}{3} = \frac{32}{3}

Therefore, the shaded area is:

323 (units2)\frac{32}{3} \, \text{(units$^2$)}

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