Question

Sketch a graph of \(y=2\) on [-1,4] and use geometry to find the exact value of \(\int_{-1}^{4} 2 d x\)

Short answer

Answer: The exact value of the integral \(\int_{-1}^{4} 2 dx\) is 10.

Step-by-step solution

Sketch the graph of y = 2 on [-1, 4]

To sketch the graph, simply draw a horizontal line at y = 2, starting from x = -1 and ending at x = 4. This is a straight line parallel to the x-axis, crossing the y-axis at y = 2.

Identify the geometric shape under the curve

In this case, the area under the curve y = 2 from x = -1 to x = 4 is a rectangle. The base of the rectangle lies on the x-axis, and it spans from x = -1 to x = 4. The height of the rectangle is equal to the value of the function, which is 2.

Calculate the area of the rectangle

To find the area of the rectangle, multiply its base length by its height. The base has a length of 5 (since 4 - (-1) = 5) and the height is 2. So, the area of the rectangle is: Area = base × height = 5 × 2 = 10

Write the exact value of the integral

Since the area under the curve is equal to the integral, we can now write the exact value of the given integral: \(\int_{-1}^{4} 2 d x = 10\) So, the exact value of the integral is 10.

Key concepts

Graph of a Function

When you have a function like \(y=2\), it's important to understand what its graph looks like. In this case, \(y=2\) is a constant function. This means it does not change as \(x\) changes. Thus, the graph is a horizontal line. Imagine a piece of graph paper, with numbers going across the bottom (the \(x\)-axis), and numbers going up the side (the \(y\)-axis). For \(y=2\), you would draw a straight line parallel to the \(x\)-axis, passing through the point where \(y\) is 2. The line continues infinitely to the left and right, but for our exercise, it starts at \(x=-1\) and ends at \(x=4\). This graphing skill is important not only for this exercise but also for understanding how functions behave visually.

• The line is horizontal because \(y\) doesn't change.
• It represents all points \((x, 2)\).
• Here, it stretches from \(x = -1\) to \(x = 4\).

Area Under a Curve

The concept of finding the area under a curve is central to understanding definite integrals. With the function \(y=2\) over the interval \([-1, 4]\), the graph forms a straight horizontal line. The space between this line and the \(x\)-axis forms a shape whose area we can calculate using geometry. Since \(y=2\) is a constant, the entire region below it forms a rectangle with a flat top. To find the area under a constant function, you essentially find the area of this rectangle. The concept of area under the curve helps to visually represent the accumulation of values, which leads directly to the integral concept.

• Look for familiar shapes like rectangles or triangles.
• The definite integral represents the total area from starting \(x\) value to ending \(x\) value.
• Here, the rectangle's base lies along the \(x\)-axis.

Rectangles in Geometry

Rectangles are a fundamental shape in geometry and are particularly useful in calculus for evaluating definite integrals. A rectangle has four sides, with opposite sides being equal and parallel. The area of a rectangle is calculated by multiplying the length of its base by its height. In the case of our exercise, the rectangle found under the curve \(y=2\) from \(x=-1\) to \(x=4\) has:

• Base: The distance from \(x=-1\) to \(x=4\), calculated as \(4 - (-1) = 5\).
• Height: This is the value of the function throughout the interval, which is 2.

To find the area, simply multiply base (5) by height (2), giving an area of 10. Knowing how to calculate areas of rectangles is crucial when determining the integral's exact value in geometric terms. This connection between geometry and integration helps in visualizing and solving many similar problems in calculus.