riemann integral from 0 to 5 of sin(x^2), n=5

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`x`²	`x`^□	log_□	√☐	^□√☐	≤	≥	□□	·	÷	`x`^◦	π
(☐)^′	`ddx`	∂∂`x`	∫	∫_□^□	lim	∑	∞	`θ`	(`f` ◦ `g`)	`f`(`x`)

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√☐

^□√☐

□□

log_□

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ddx

≥	≤	·	÷	`x`^◦	(☐)	\|☐\|	(`f` ◦ `g`)	`f`(`x`)	ln	`e`^☐
(☐)^′	∂∂`x`	∫_□^□	lim	∑	sin	cos	tan	cot	csc	sec

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Steps Examples

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∫₀⁵sin(x²)dx, n=5

Solution

1.66670…

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Solve by:

Solve by midpoint Riemann sum

Solve by left point Riemann sum

Solve by right point Riemann sum

Solve by trapezoidal rule

Solve by Simpson's rule

One step at a time

Riemann sum using midpoint of subinterval

∫_a^bf(x)dx ≈ Δ x(f(x₀+x₁2 )+f(x₁+x₂2 )+f(x₂+x₃2 )+...+f(x_n−1+x_n2 )),
where Δ x = b−an

Apply Riemann Formula

=1· (f(x₀+x₁2 )+f(x₁+x₂2 )+f(x₂+x₃2 )+f(x₃+x₄2 )+f(x₄+x₅2 ))

Calculate Sub Intervals

=1· (sin(14 )+sin(94 )+sin(254 )+sin(494 )+sin(814 ))

=1.66670…

Midpoint Rule Examples

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Giải quyết bằng cách: left point Riemann sum

Giải quyết bằng cách: right point Riemann sum

Giải quyết bằng cách: trapezoidal rule

Giải quyết bằng cách: Simpson's rule

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Notebook

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−▭▭	<	7	8	9	÷	`AC`
+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

riemann integral from 0 to 5 of sin(x^2), n=5

Solution

Solution steps

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