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?

Answer 1

# int_(pi/2)^pisqrt(x)sinxdx ~~ 1.430 # (3dp)

The values of y are tabulated as follows (using Excel)

Using the trapezoidal rule:
# int_a^bydx ~~ h/2{(y_0+y_n)+2(y_1+y_2+...+y_(n-1))}#

we have:
# int_(pi/2)^pisqrt(x)sinxdx ~~ (pi/8)/2{ (1.25331 + 0.00000) + 2(1.29458 + 1.08540 + 0.63448)} #
# :. int_(pi/2)^pisqrt(x)sinxdx ~~ pi/16{ 1.25331 + 2(3.01446)} #
# :. int_(pi/2)^pisqrt(x)sinxdx ~~ pi/16 { 1.25331 + 6.02892} #
# :. int_(pi/2)^pisqrt(x)sinxdx ~~ pi/16 { 7.28223} #
# :. int_(pi/2)^pisqrt(x)sinxdx ~~ 1.42986 #
# :. int_(pi/2)^pisqrt(x)sinxdx ~~ 1.430 # (3dp)

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Answer 2

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.

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

  2. 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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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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