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Exam-Style Question on Integration

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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:

$$ R_n = \int^{x_{n+1}}_{x_n} y \; dx$$ Graph Regions

(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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