How do you use the trapezoidal rule with n=4 to approximate the area between the curve #sqrt(x) sinx# from pi/2 to pi?
The values of y are tabulated as follows (using Excel)
Using the trapezoidal rule:
we have:
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To approximate the area between the curve ( \sqrt{x} \sin(x) ) from ( \frac{\pi}{2} ) to ( \pi ) using the trapezoidal rule with ( n = 4 ), we first need to calculate the interval ( \Delta x ) and then apply the trapezoidal rule formula.

Calculate ( \Delta x ): ( \Delta x = \frac{b  a}{n} = \frac{\pi  \frac{\pi}{2}}{4} = \frac{\pi}{4} ).

Apply the trapezoidal rule formula for ( n = 4 ): ( \text{Area} \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)] ).
Substitute the values: ( \text{Area} \approx \frac{\pi}{4 \times 2} [\sqrt{\frac{\pi}{2}}\sin\frac{\pi}{2} + 2\sqrt{\frac{5\pi}{4}}\sin\frac{5\pi}{4} + 2\sqrt{\frac{3\pi}{2}}\sin\frac{3\pi}{2} + 2\sqrt{\frac{7\pi}{4}}\sin\frac{7\pi}{4} + \sqrt{\pi}\sin\pi] ).
Simplify the expression to find the approximate area.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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