You might need:
Calculator
Approximate the area between $h(x)$ and the $x$-axis from $x=-2$ to $x=4$ using a trapezoidal sum with 3 equal subdivisions.
$$
T(3)=
$$
$\square$

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To approximate the area between $h(x)$ and the $x$-axis from $x = -2$ to $x = 4$ using a trapezoidal sum with 3 equal subdivisions, we first determine the width of each subdivision. The interval from $-2$ to $4$ is $6$ units long, so each subdivision has a width of $2$ units. ‖ ‖ step 2 ⋮ The trapezoidal rule formula for $n$ subdivisions is given by: $$ T(n) = \frac{b - a}{2n} \left[ h(x_0) + 2 \sum_{i=1}^{n-1} h(x_i) + h(x_n) \right] $$ For $n = 3$, $a = -2$, and $b = 4$, the formula becomes: $$ T(3) = \frac{4 - (-2)}{2 \cdot 3} \left[ h(-2) + 2(h(0) + h(2)) + h(4) \right] $$ ‖ ‖ step 3 ⋮ Calculate the width of each subdivision: $$ \Delta x = \frac{4 - (-2)}{3} = 2 $$ ‖ ‖ step 4 ⋮ Identify the $x$ values at the endpoints and midpoints of the subdivisions: $$ x_0 = -2, \quad x_1 = 0, \quad x_2 = 2, \quad x_3 = 4 $$ ‖ ‖ step 5 ⋮ Substitute the $x$ values into the trapezoidal rule formula: $$ T(3) = \frac{6}{6} \left[ h(-2) + 2(h(0) + h(2)) + h(4) \right] $$ $$ T(3) = 1 \left[ h(-2) + 2(h(0) + h(2)) + h(4) \right] $$ ‖ ‖ step 6 ⋮ Using the given graph, approximate the values of $h(x)$ at the specified points: $$ h(-2) \approx 0, \quad h(0) \approx 4, \quad h(2) \approx 6, \quad h(4) \approx 2 $$ ‖ ‖ step 7 ⋮ Substitute these values into the formula: $$ T(3) = 1 \left[ 0 + 2(4 + 6) + 2 \right] $$ $$ T(3) = 1 \left[ 0 + 2(10) + 2 \right] $$ $$ T(3) = 1 \left[ 0 + 20 + 2 \right] $$ $$ T(3) = 22 $$ ‖ ∻Answer∻ ⚹ The approximate area between $h(x)$ and the $x$-axis from $x = -2$ to $x = 4$ using a trapezoidal sum with 3 equal subdivisions is $22$ square units. ⚹ ∻Key Concept∻ ⚹Trapezoidal Rule⚹ ∻Explanation∻ ⚹The trapezoidal rule is a numerical method to approximate the definite integral of a function. It works by dividing the area under the curve into trapezoids, calculating the area of each, and summing them up.⚹◊What is the formula for calculating the area between a function and the x-axis using a trapezoidal sum with 3 equal subdivisions?⍭ Generate me a similar question◊

Sia