 # Exam-Style Question on Integration

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 661. This question is similar to one that appeared on an IB AA Higher paper in 2023. The use of a calculator is allowed.

The curve $$y=\sin(\sqrt{x}) \text{ where } x \ge 0$$ intersects the x axis at the points $$x_0, x_1, x_2, x_3, x_4, ...$$ and $$x_0 = 0$$.

(a) Find $$x_1$$, the first x-intercept of the function to the right of the origin. Give your answer in terms of $$\pi$$.

(b) Find an expression for the nth x-intercept of the curve, in terms of $$\pi$$.

(c) By using an appropriate substitution, show that:

$$\int \sin(\sqrt{x}) \; dx = 2\sin(\sqrt{x}) - 2 \sqrt{x} \cos(\sqrt{x})$$

The area of the region bounded by the curve and the x-axis is denoted by $$R_n$$ where:

$$R_n = \int^{x_{n+1}}_{x_n} y \; dx$$ (d) Calculate the area of region $$R_n$$ giving your answer in terms of $$\pi$$.

(e) Hence, show that the areas of the regions bounded by the curve and the x-axis, form an arithmetic sequence.

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