Areas and further applications Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Areas and further applications questions. See exactly how to solve problems on areas, definite-integral, set-up, identify.

areasdefinite-integralset-upidentifyrootsbetween-curves
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Find the area of the region bounded by the curve y=x2y=x^{2}, the xx-axis and the lines x=0x=0 and x=3x=3.

Worked solution

  1. Write the area as a definite integral

    A=∫03x2 dxA=\int_{0}^{3} x^{2}\,dx

    The area under a curve above the x-axis is a definite integral of y.

  2. Integrate and evaluate between the limits

    A=[x33]03=9A=\left[\frac{x^{3}}{3}\right]_{0}^{3}=9

    Substituting the limits into the antiderivative gives the area.

  3. State the exact area

    A=9A=9

    This is the required area.

Answer
99
Question 2
2 markseasy
Find the area of the region bounded by the curve y=x2y=x^{2}, the xx-axis and the lines x=0x=0 and x=2x=2.

Worked solution

  1. Write the area as a definite integral

    A=∫02x2 dxA=\int_{0}^{2} x^{2}\,dx

    The area under a curve above the x-axis is a definite integral of y.

  2. Integrate and evaluate between the limits

    A=[x33]02=83A=\left[\frac{x^{3}}{3}\right]_{0}^{2}=\frac{8}{3}

    Substituting the limits into the antiderivative gives the area.

  3. State the exact area

    A=83A=\frac{8}{3}

    This is the required area.

Answer
83\frac{8}{3}
Question 3
2 markseasy
Find the area of the region bounded by the curve y=2xy=2 x, the xx-axis and the lines x=0x=0 and x=4x=4.

Worked solution

  1. Write the area as a definite integral

    A=∫042x dxA=\int_{0}^{4} 2 x\,dx

    The area under a curve above the x-axis is a definite integral of y.

  2. Integrate and evaluate between the limits

    A=[x2]04=16A=\left[x^{2}\right]_{0}^{4}=16

    Substituting the limits into the antiderivative gives the area.

  3. State the exact area

    A=16A=16

    This is the required area.

Answer
1616
Question 4
2 markseasy
Find the area of the region bounded by the curve y=x2+1y=x^{2} + 1, the xx-axis and the lines x=0x=0 and x=2x=2.

Worked solution

  1. Write the area as a definite integral

    A=∫02x2+1 dxA=\int_{0}^{2} x^{2} + 1\,dx

    The area under a curve above the x-axis is a definite integral of y.

  2. Integrate and evaluate between the limits

    A=[x33+x]02=143A=\left[\frac{x^{3}}{3} + x\right]_{0}^{2}=\frac{14}{3}

    Substituting the limits into the antiderivative gives the area.

  3. State the exact area

    A=143A=\frac{14}{3}

    This is the required area.

Answer
143\frac{14}{3}
Question 5
2 markseasy
Find the area of the region bounded by the curve y=3x2y=3 x^{2}, the xx-axis and the lines x=1x=1 and x=2x=2.

Worked solution

  1. Write the area as a definite integral

    A=∫123x2 dxA=\int_{1}^{2} 3 x^{2}\,dx

    The area under a curve above the x-axis is a definite integral of y.

  2. Integrate and evaluate between the limits

    A=[x3]12=7A=\left[x^{3}\right]_{1}^{2}=7

    Substituting the limits into the antiderivative gives the area.

  3. State the exact area

    A=7A=7

    This is the required area.

Answer
77

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