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Understanding the trapezoidal rule

AP.CALC:
LIM‑5 (EU)
,
LIM‑5.A (LO)
,
LIM‑5.A.1 (EK)
,
LIM‑5.A.2 (EK)
,
LIM‑5.A.3 (EK)
,
LIM‑5.A.4 (EK)
Walk through an example using the trapezoid rule, then try a couple of practice problems on your own.
By now you know that we can use Riemann sums to approximate the area under a function. Riemann sums use rectangles, which make for some pretty sloppy approximations. But what if we used trapezoids to approximate the area under a function instead?
Key idea: By using trapezoids (aka the "trapezoid rule") we can get more accurate approximations than by using rectangles (aka "Riemann sums").

An example of the trapezoid rule

Let's check it out by using three trapezoids to approximate the area under the function f, left parenthesis, x, right parenthesis, equals, 3, natural log, left parenthesis, x, right parenthesis on the interval open bracket, 2, comma, 8, close bracket.
Here's how that looks in a diagram when we call the first trapezoid T, start subscript, 1, end subscript, the second trapezoid T, start subscript, 2, end subscript, and the third trapezoid T, start subscript, 3, end subscript:
Recall that the area of a trapezoid is h, left parenthesis, start fraction, b, start subscript, 1, end subscript, plus, b, start subscript, 2, end subscript, divided by, 2, end fraction, right parenthesis where h is the height and b, start subscript, 1, end subscript and b, start subscript, 2, end subscript are the bases.

Finding the area of T, start subscript, 1, end subscript

We need to think about the trapezoid as if it's lying sideways.
The height h is the 2 at the bottom of T, start subscript, 1, end subscript that spans x, equals, start color #1fab54, 2, end color #1fab54 to x, equals, start color #ca337c, 4, end color #ca337c.
The first base b, start subscript, 1, end subscript is the value of 3, natural log, left parenthesis, x, right parenthesis at x, equals, start color #1fab54, 2, end color #1fab54, which is 3, natural log, left parenthesis, start color #1fab54, 2, end color #1fab54, right parenthesis.
The second base b, start subscript, 2, end subscript is the value of 3, natural log, left parenthesis, x, right parenthesis at x, equals, start color #ca337c, 4, end color #ca337c, which is 3, natural log, left parenthesis, start color #ca337c, 4, end color #ca337c, right parenthesis.
Here's how all of this looks visually:
Let's put this all together to find the area of T, start subscript, 1, end subscript:
T, start subscript, 1, end subscript, equals, h, left parenthesis, start fraction, b, start subscript, 1, end subscript, plus, b, start subscript, 2, end subscript, divided by, 2, end fraction, right parenthesis
T, start subscript, 1, end subscript, equals, 2, left parenthesis, start fraction, 3, natural log, left parenthesis, start color #1fab54, 2, end color #1fab54, right parenthesis, plus, 3, natural log, left parenthesis, start color #ca337c, 4, end color #ca337c, right parenthesis, divided by, 2, end fraction, right parenthesis
Simplify:
T, start subscript, 1, end subscript, equals, 3, left parenthesis, natural log, left parenthesis, start color #1fab54, 2, end color #1fab54, right parenthesis, plus, natural log, left parenthesis, start color #ca337c, 4, end color #ca337c, right parenthesis, right parenthesis

Finding the area of T, start subscript, 2, end subscript

Let's find the height and both of the bases:
h, equals, 2
b, start subscript, 1, end subscript, equals, 3, natural log, left parenthesis, 4, right parenthesis
b, start subscript, 2, end subscript, equals, 3, natural log, left parenthesis, 6, right parenthesis
Plug in and simplify:
T, start subscript, 2, end subscript, equals, 3, left parenthesis, natural log, left parenthesis, 4, right parenthesis, plus, natural log, left parenthesis, 6, right parenthesis, right parenthesis

Find the area of T, start subscript, 3, end subscript

T, start subscript, 3, end subscript, equals
Choose 1 answer:
Choose 1 answer:

Finding the total area approximation

We find the total area by adding up the area of each of the three trapezoids:
start text, T, o, t, a, l, space, a, r, e, a, end text, equals, T, start subscript, 1, end subscript, plus, T, start subscript, 2, end subscript, plus, T, start subscript, 3, end subscript
Here's the final simplified answer:
start text, T, o, t, a, l, space, a, r, e, a, end text, equals, 3, left parenthesis, natural log, 2, plus, 2, natural log, 4, plus, 2, natural log, 6, plus, natural log, 8, right parenthesis
You should pause here and walk through the algebra to make sure you understand how we got this!

Practice problem

Choose the expression that uses four trapezoids to approximate the area under the function f, left parenthesis, x, right parenthesis, equals, 2, natural log, left parenthesis, x, right parenthesis on the interval open bracket, 2, comma, 8, close bracket.
Choose 1 answer:
Choose 1 answer:

Challenge problem

Choose the expression that uses three trapezoids to approximate the area under the function f on the interval open bracket, minus, 1, comma, 5, close bracket.
Choose 1 answer:
Choose 1 answer:

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