Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(d) ∫₀⁷ ƒ(𝓍) d𝓍

Verified step by step guidance

Step 1: Observe the graph of the function ƒ(x) provided. The graph is piecewise linear, meaning it consists of straight-line segments. To evaluate the definite integral ∫₀⁷ ƒ(x) dx using geometry, we need to calculate the areas of the geometric shapes formed between the graph and the x-axis over the interval [0, 7].

Step 2: Break the interval [0, 7] into subintervals based on where the function changes behavior. From the graph, the function changes at x = 0, x = 3, x = 5, and x = 7. Identify the shapes formed in each subinterval: rectangles and triangles.

Step 3: Calculate the area of each shape. For the interval [0, 3], the graph forms a rectangle with height 2 and width 3. For the interval [3, 5], the graph forms a trapezoid (or two triangles stacked) with heights 2 and 3 and width 2. For the interval [5, 7], the graph forms a triangle below the x-axis with base 2 and height -3.

Step 4: Use the formula for the area of a rectangle (Area = base × height) and the formula for the area of a triangle (Area = 0.5 × base × height) to compute the areas of each shape. Remember to account for the sign of the area: areas above the x-axis are positive, and areas below the x-axis are negative.

Step 5: Add the areas of all the shapes together to find the total area under the curve from x = 0 to x = 7. This sum represents the value of the definite integral ∫₀⁷ ƒ(x) dx.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral represents the signed area under a curve between two points on the x-axis. It is calculated using the limits of integration, which define the interval over which the area is measured. In this context, the definite integral ∫₀⁷ ƒ(𝓍) d𝓍 calculates the total area between the graph of the function ƒ and the x-axis from x = 0 to x = 7.

Area Under the Curve

The area under the curve of a function can be interpreted geometrically as the total area between the curve and the x-axis. This area can be positive or negative depending on whether the curve is above or below the x-axis. In the given problem, the graph shows a piecewise linear function, which allows for straightforward geometric calculations of the areas of rectangles and triangles.

Piecewise Function

A piecewise function is defined by different expressions or formulas over different intervals of its domain. In this case, the function ƒ(𝓍) is represented by linear segments across specified intervals. Understanding how to evaluate the function at different segments is crucial for accurately calculating the definite integral, as each segment contributes differently to the total area.

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Piecewise Functions

Related Practice

Textbook Question

Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 𝓍⁴ ― 𝓍² on [―1, 1]

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Textbook Question

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t if ―2 ≤ t < 0

t²/2 if 0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(a) Evaluate F(―2) and F(2).

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Textbook Question

Evaluating integrals Evaluate the following integrals.