Area Under a Curve (Year 11-12)
Free printable Year 11-12 Mathematical Methods worksheet: using integration to find the exact area between a polynomial curve and the x-axis.
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Year 11-12 · Math worksheet
Name
Date
Math
Integration: Area Under a Curve
For each curve (which stays above the x-axis on the given interval), find the exact area of the shaded region between the curve and the x-axis by evaluating the definite integral from a to b.
- 1.Find the exact area of the shaded region between the curve y = 3x^2 + 5x + 5 and the x-axis, from x = 0 to x = 3. (The curve stays above the x-axis on this interval.)
y = 3x^2 + 5x + 5: shaded area from x = 0 to x = 3. - 2.Find the exact area of the shaded region between the curve y = 4x^2 + 2x + 3 and the x-axis, from x = 0 to x = 2. (The curve stays above the x-axis on this interval.)
y = 4x^2 + 2x + 3: shaded area from x = 0 to x = 2. - 3.Find the exact area of the shaded region between the curve y = 3x^2 + 4x + 1 and the x-axis, from x = 0 to x = 3. (The curve stays above the x-axis on this interval.)
y = 3x^2 + 4x + 1: shaded area from x = 0 to x = 3. - 4.Find the exact area of the shaded region between the curve y = x + 1 and the x-axis, from x = 0 to x = 1. (The curve stays above the x-axis on this interval.)
y = x + 1: shaded area from x = 0 to x = 1. - 5.Find the exact area of the shaded region between the curve y = 3x^3 + + 5x + 2 and the x-axis, from x = 2 to x = 5. (The curve stays above the x-axis on this interval.)
y = 3x^3 + + 5x + 2: shaded area from x = 2 to x = 5. - 6.Find the exact area of the shaded region between the curve y = 4x^2 + 4x + 1 and the x-axis, from x = 2 to x = 3. (The curve stays above the x-axis on this interval.)
y = 4x^2 + 4x + 1: shaded area from x = 2 to x = 3. - 7.Find the exact area of the shaded region between the curve y = 2x + 3 and the x-axis, from x = 1 to x = 4. (The curve stays above the x-axis on this interval.)
y = 2x + 3: shaded area from x = 1 to x = 4. - 8.Find the exact area of the shaded region between the curve y = x + 5 and the x-axis, from x = 0 to x = 3. (The curve stays above the x-axis on this interval.)
y = x + 5: shaded area from x = 0 to x = 3.
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About this worksheet
In class: For a curve that stays above the x-axis, the area between it and the axis from a to b is exactly the definite integral. Sketch the region first, then evaluate F(b) - F(a). Give the area in square units.
This Year 11-12 math set is free to print or save as a PDF, and its answer key is computed in code, so it is never wrong.