1.11.5 Integration by substitution and by parts (A-level only)

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Question 58

Using the substitution $\frac{3}{2}\sin 2x$ , show that

\int_0^{\frac{\pi}{2}} e^{3x + \frac{3}{2}\sin 2x} \cos^2 x \, dx = \frac{1}{6}(e^{\frac{3\pi}{2}} - 1)

[4]

The design of a high-performance aerodynamic component involves a surface generated by rotating a region $R$ through $2π2\pi$ radians about the $x$ -axis. The region $R$ is bounded by the curve with equation

\sqrt{18} e^{\frac{3}{2}x + \frac{3}{4}\sin 2x} \cos x

and the coordinate axes in the first quadrant.

Use the result from part (a) to find the volume of the component, giving your answer in simplest form.

[3]

Using the substitution $\frac{3}{2}\sin 2x$ , show that

\int_0^{\frac{\pi}{2}} e^{3x + \frac{3}{2}\sin 2x} \cos^2 x \, dx = \frac{1}{6}(e^{\frac{3\pi}{2}} - 1)

[4]

\sqrt{18} e^{\frac{3}{2}x + \frac{3}{4}\sin 2x} \cos x

and the coordinate axes in the first quadrant.

Use the result from part (a) to find the volume of the component, giving your answer in simplest form.

1.11.5 Integration by substitution and by parts (A-level only)

Question 58

Markscheme

1.11.5 Integration by substitution and by parts (A-level only)

Question 58

Markscheme

1.11.5 Integration by substitution and by parts (A-level only)

Question 58

1.11.5 Integration by substitution and by parts (A-level only) Questions

1.11.5 Integration by substitution and by parts (A-level only)

Question 58

1.11.5 Integration by substitution and by parts (A-level only) Questions