Math / Geometry

QuestionEvaluate the following integrals by interpreting them in terms of areas: (a) $\int_{0}^{2} f(x) d x=$ $\square$ 1 s

Studdy Solution

STEP 1

1. The function $f(x)$ is piecewise linear and defined from $x = 0$ to $x = 2$ .
2. The graph of $f(x)$ forms a right triangle with the x-axis from $x = 0$ to $x = 2$ .
3. The vertices of the triangle are at points $(0,0)$ , $(2,2)$ , and $(2,0)$ .

STEP 2

1. Identify the geometric shape formed by the graph of $f(x)$ and the x-axis.
2. Calculate the area of the geometric shape.
3. Interpret the integral as the area under the curve.

STEP 3

Identify the geometric shape formed by the graph of $f(x)$ and the x-axis. The graph forms a right triangle with the x-axis, with vertices at $(0,0)$ , $(2,2)$ , and $(2,0)$ .

STEP 4

Calculate the area of the right triangle. The base of the triangle is along the x-axis from $x = 0$ to $x = 2$ , so the base length is $2$ . The height of the triangle is the y-coordinate at $x = 2$ , which is $2$ .

STEP 5

Use the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Substitute the values for base and height: $\text{Area} = \frac{1}{2} \times 2 \times 2$

STEP 6

Calculate the area: $\text{Area} = \frac{1}{2} \times 2 \times 2 = 2$

STEP 7

Interpret the integral $\int_{0}^{2} f(x) \, dx$ as the area under the curve from $x = 0$ to $x = 2$ . Since this area is the area of the triangle calculated, the value of the integral is:
$\boxed{2}$

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QuestionEvaluate the following integrals by interpreting them in terms of areas: (a) $\int_{0}^{2} f(x) d x=$ $\square$ 1 s

Studdy Solution

STEP 1

1. The function $f(x)$ is piecewise linear and defined from $x = 0$ to $x = 2$ .
2. The graph of $f(x)$ forms a right triangle with the x-axis from $x = 0$ to $x = 2$ .
3. The vertices of the triangle are at points $(0,0)$ , $(2,2)$ , and $(2,0)$ .

STEP 2

1. Identify the geometric shape formed by the graph of $f(x)$ and the x-axis.
2. Calculate the area of the geometric shape.
3. Interpret the integral as the area under the curve.

STEP 3

Identify the geometric shape formed by the graph of $f(x)$ and the x-axis. The graph forms a right triangle with the x-axis, with vertices at $(0,0)$ , $(2,2)$ , and $(2,0)$ .

STEP 4

Calculate the area of the right triangle. The base of the triangle is along the x-axis from $x = 0$ to $x = 2$ , so the base length is $2$ . The height of the triangle is the y-coordinate at $x = 2$ , which is $2$ .

STEP 5

Use the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Substitute the values for base and height: $\text{Area} = \frac{1}{2} \times 2 \times 2$

STEP 6

Calculate the area: $\text{Area} = \frac{1}{2} \times 2 \times 2 = 2$

STEP 7

Interpret the integral $\int_{0}^{2} f(x) \, dx$ as the area under the curve from $x = 0$ to $x = 2$ . Since this area is the area of the triangle calculated, the value of the integral is:
$\boxed{2}$

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QuestionEvaluate the following integrals by interpreting them in terms of areas: (a) ∫02f(x)dx=\int_{0}^{2} f(x) d x=∫02​f(x)dx= □\square□ 1 s

Studdy Solution

STEP 1

STEP 2

1. Identify the geometric shape formed by the graph of f(x) f(x) f(x) and the x-axis.2. Calculate the area of the geometric shape.3. Interpret the integral as the area under the curve.

STEP 3

Identify the geometric shape formed by the graph of f(x) f(x) f(x) and the x-axis. The graph forms a right triangle with the x-axis, with vertices at (0,0) (0,0) (0,0), (2,2) (2,2) (2,2), and (2,0) (2,0) (2,0).

STEP 4

Calculate the area of the right triangle. The base of the triangle is along the x-axis from x=0 x = 0 x=0 to x=2 x = 2 x=2, so the base length is 2 2 2. The height of the triangle is the y-coordinate at x=2 x = 2 x=2, which is 2 2 2.

STEP 5

Use the formula for the area of a triangle: Area=12×base×height \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} Area=21​×base×height Substitute the values for base and height: Area=12×2×2 \text{Area} = \frac{1}{2} \times 2 \times 2 Area=21​×2×2

STEP 6

Calculate the area: Area=12×2×2=2 \text{Area} = \frac{1}{2} \times 2 \times 2 = 2 Area=21​×2×2=2

STEP 7

Interpret the integral ∫02f(x) dx\int_{0}^{2} f(x) \, dx∫02​f(x)dx as the area under the curve from x=0 x = 0 x=0 to x=2 x = 2 x=2. Since this area is the area of the triangle calculated, the value of the integral is:2 \boxed{2} 2​

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QuestionEvaluate the following integrals by interpreting them in terms of areas: (a) $\int_{0}^{2} f(x) d x=$ $\square$ 1 s

1. Identify the geometric shape formed by the graph of $f(x)$ and the x-axis.
2. Calculate the area of the geometric shape.
3. Interpret the integral as the area under the curve.

Identify the geometric shape formed by the graph of $f(x)$ and the x-axis. The graph forms a right triangle with the x-axis, with vertices at $(0,0)$ , $(2,2)$ , and $(2,0)$ .

Calculate the area of the right triangle. The base of the triangle is along the x-axis from $x = 0$ to $x = 2$ , so the base length is $2$ . The height of the triangle is the y-coordinate at $x = 2$ , which is $2$ .

Use the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Substitute the values for base and height: $\text{Area} = \frac{1}{2} \times 2 \times 2$

Calculate the area: $\text{Area} = \frac{1}{2} \times 2 \times 2 = 2$

Interpret the integral $\int_{0}^{2} f(x) \, dx$ as the area under the curve from $x = 0$ to $x = 2$ . Since this area is the area of the triangle calculated, the value of the integral is:
$\boxed{2}$