Further integration techniques Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Further integration techniques questions. See exactly how to solve problems on arc-length, cartesian-form, integration, parametric-form.

arc-lengthcartesian-formintegrationparametric-formpolar-formsurface-of-revolution
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
The curve CC has equation y=cosh⁡(x)y=\cosh{\left(x \right)}. Find the exact length of the arc of CC from x=0x=0 to x=1x=1.

Worked solution

  1. State the arc-length formula for a cartesian curve

    s=∫011+(dydx)2 dxs=\int_{0}^{1}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx

    The limits are the xx-coordinates of the two ends of the arc.

  2. Differentiate the equation of the curve

    dydx=sinh⁡(x)\frac{dy}{dx}=\sinh{\left(x \right)}

    The gradient is what feeds into the arc-length integrand.

  3. Square the gradient and add one

    1+(dydx)2=cosh⁡2(x)1+\left(\frac{dy}{dx}\right)^{2}=\cosh^{2}{\left(x \right)}

    This expression must be a perfect square if the integral is to be exact.

  4. State the exact arc length

    s=sinh⁡(1)s=\sinh{\left(1 \right)}

    This is the exact length of the arc, left in exact form.

Answer
sinh⁡(1)\sinh{\left(1 \right)}
Question 2
2 markseasy
The curve CC has equation y=3cosh⁡(x3)y=3 \cosh{\left(\frac{x}{3} \right)}. Find the exact length of the arc of CC from x=0x=0 to x=3x=3.

Worked solution

  1. State the arc-length formula for a cartesian curve

    s=∫031+(dydx)2 dxs=\int_{0}^{3}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx

    The limits are the xx-coordinates of the two ends of the arc.

  2. Differentiate the equation of the curve

    dydx=sinh⁡(x3)\frac{dy}{dx}=\sinh{\left(\frac{x}{3} \right)}

    The gradient is what feeds into the arc-length integrand.

  3. Square the gradient and add one

    1+(dydx)2=cosh⁡2(x3)1+\left(\frac{dy}{dx}\right)^{2}=\cosh^{2}{\left(\frac{x}{3} \right)}

    This expression must be a perfect square if the integral is to be exact.

  4. State the exact arc length

    s=3sinh⁡(1)s=3 \sinh{\left(1 \right)}

    This is the exact length of the arc, left in exact form.

Answer
3sinh⁡(1)3 \sinh{\left(1 \right)}
Question 3
2 markseasy
The curve CC has equation y=2x323y=\frac{2 x^{\frac{3}{2}}}{3}. Find the exact length of the arc of CC from x=0x=0 to x=3x=3.

Worked solution

  1. State the arc-length formula for a cartesian curve

    s=∫031+(dydx)2 dxs=\int_{0}^{3}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx

    The limits are the xx-coordinates of the two ends of the arc.

  2. Differentiate the equation of the curve

    dydx=x\frac{dy}{dx}=\sqrt{x}

    The gradient is what feeds into the arc-length integrand.

  3. Square the gradient and add one

    1+(dydx)2=x+11+\left(\frac{dy}{dx}\right)^{2}=x + 1

    This expression must be a perfect square if the integral is to be exact.

  4. State the exact arc length

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

    This is the exact length of the arc, left in exact form.

Answer
143\frac{14}{3}
Question 4
2 markseasy
The curve CC has equation y=(x2+2)323y=\frac{\left(x^{2} + 2\right)^{\frac{3}{2}}}{3}. Find the exact length of the arc of CC from x=0x=0 to x=3x=3.

Worked solution

  1. State the arc-length formula for a cartesian curve

    s=∫031+(dydx)2 dxs=\int_{0}^{3}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx

    The limits are the xx-coordinates of the two ends of the arc.

  2. Differentiate the equation of the curve

    dydx=xx2+2\frac{dy}{dx}=x \sqrt{x^{2} + 2}

    The gradient is what feeds into the arc-length integrand.

  3. Square the gradient and add one

    1+(dydx)2=(x2+1)21+\left(\frac{dy}{dx}\right)^{2}=\left(x^{2} + 1\right)^{2}

    This expression must be a perfect square if the integral is to be exact.

  4. State the exact arc length

    s=12s=12

    This is the exact length of the arc, left in exact form.

Answer
1212
Question 5
2 markseasy
A curve has parametric equations x=2cos⁡(t)x=2 \cos{\left(t \right)}, y=2sin⁡(t)y=2 \sin{\left(t \right)}, for 0≤t≤π30\le t\le \frac{\pi}{3}. Find the exact length of the curve.

Worked solution

  1. State the arc-length formula for a parametric curve

    s=∫0π3(dxdt)2+(dydt)2 dts=\int_{0}^{\frac{\pi}{3}}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt

    The limits are the parameter values at the two ends of the curve.

  2. Differentiate both parametric equations

    dxdt=−2sin⁡(t),dydt=2cos⁡(t)\frac{dx}{dt}=- 2 \sin{\left(t \right)},\qquad\frac{dy}{dt}=2 \cos{\left(t \right)}

    Both derivatives are needed for the speed of the point along the curve.

  3. Form the sum of the squares

    (dxdt)2+(dydt)2=4\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}=4

    Simplifying this sum is the key step; it must reduce to a perfect square.

  4. State the exact length of the curve

    s=2π3s=\frac{2 \pi}{3}

    This is the exact length of the curve over the given parameter range.

Answer
2π3\frac{2 \pi}{3}

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