Math / Calculus

Question13 14 12.5 13.3333

Studdy Solution

STEP 1

What is this asking? We need to find the area under a curve using trapezoids, like building a funky staircase with slanted steps! Watch out! Don't forget to divide the interval into the correct number of trapezoids, and make sure you're using the right formula for the area of a trapezoid!

STEP 2

1. Define the function and interval
2. Calculate the width of each trapezoid
3. Calculate the heights of the trapezoids
4. Calculate the area of each trapezoid
5. Sum the areas

STEP 3

We're given the function $f(x) = \frac{x^2}{2} - x + 1$ and the interval $[-3, 1]$ .
This means we're looking at the area under the curve between $x = -3$ and $x = 1$ .

STEP 4

We're using $n = \textbf{4}$ trapezoids.
The width of each trapezoid, which we'll call $\Delta x$ , is calculated by dividing the length of the interval by the number of trapezoids.

STEP 5

The interval length is $1 - (-3) = \textbf{4}$ .

STEP 6

So, $\Delta x = \frac{4}{4} = \textbf{1}$ .
Each trapezoid has a width of 1!

STEP 7

The heights of the trapezoids are determined by the function's values at the endpoints of each trapezoid.
Since our trapezoids have a width of 1, and we start at $x = \textbf{-3}$ , our $x$ values are $-3, -2, -1, 0, 1$ .

STEP 8

Let's plug these $x$ values into our function $f(x)$ to get the heights: $f(-3) = \frac{(-3)^2}{2} - (-3) + 1 = \frac{9}{2} + 3 + 1 = 4.5 + 3 + 1 = \textbf{8.5}$ $f(-2) = \frac{(-2)^2}{2} - (-2) + 1 = \frac{4}{2} + 2 + 1 = 2 + 2 + 1 = \textbf{5}$ $f(-1) = \frac{(-1)^2}{2} - (-1) + 1 = \frac{1}{2} + 1 + 1 = 0.5 + 1 + 1 = \textbf{2.5}$ $f(0) = \frac{0^2}{2} - 0 + 1 = 0 - 0 + 1 = \textbf{1}$ $f(1) = \frac{1^2}{2} - 1 + 1 = \frac{1}{2} - 1 + 1 = \textbf{0.5}$

STEP 9

Remember, the area of a trapezoid is $\frac{1}{2} \cdot \text{width} \cdot (\text{height}_1 + \text{height}_2)$ .
Since our width is 1, the area of each trapezoid simplifies to $\frac{1}{2} (\text{height}_1 + \text{height}_2)$ .

STEP 10

Area 1: $\frac{1}{2}(8.5 + 5) = \frac{13.5}{2} = \textbf{6.75}$ Area 2: $\frac{1}{2}(5 + 2.5) = \frac{7.5}{2} = \textbf{3.75}$ Area 3: $\frac{1}{2}(2.5 + 1) = \frac{3.5}{2} = \textbf{1.75}$ Area 4: $\frac{1}{2}(1 + 0.5) = \frac{1.5}{2} = \textbf{0.75}$

STEP 11

Now, we just add up the areas of all the trapezoids to get the approximate area under the curve: $6.75 + 3.75 + 1.75 + 0.75 = \textbf{13}$ .

STEP 12

The approximate area under the curve $f(x) = \frac{x^2}{2} - x + 1$ on the interval $[-3, 1]$ using 4 trapezoids is 13.

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