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Magazine Chapter 5: Problem 26
Use known area formulas to evaluate the integrals in Exercises \(23-28\) $$ \int_{a}^{b} 3 t d t, \quad 0< a < b $$
Short Answer
Expert verified
\( \frac{3}{2} (b^2 - a^2) \)
Step by step solution
01
Identify the Integral
We need to evaluate the integral \( \int_{a}^{b} 3t \, dt \) over the interval \( (a, b) \) with the condition \( 0 < a < b \).
02
Understand the Geometric Interpretation
The function \( 3t \) represents a straight line, and the integral of this function from \( a \) to \( b \) corresponds to the area under the curve from \( a \) to \( b \). This forms a trapezoid or a 'curved quadrilateral.'
03
Area of Trapezoid Formula
The area \( A \) of a trapezoid is given by the formula \[ A = \frac{1}{2} (b_1 + b_2) \times h \], where \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides, and \( h \) is the distance (height) between them.
04
Apply the Area Formula to Our Integral
Here, \( b_1 = 3a \) and \( b_2 = 3b \) are the lengths of the parallel sides (when \( x = a \) and \( x = b \), the line has heights \( 3a \) and \( 3b \) respectively). The height \( h = b - a \).
05
Compute the Area
Substitute these values into the formula: \[ A = \frac{1}{2} (3a + 3b)(b-a) = \frac{1}{2} \cdot 3(a+b)(b-a) = \frac{3}{2}(a+b)(b-a) \].
06
Simplify the Final Answer
The expression \( \frac{3}{2}(a+b)(b-a) \) simplifies to \[ \frac{3}{2} (b^2 - a^2) \] using the distributive property.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Area Formula
When we deal with integrals, especially in a geometric context, we often think about area. Integrals can help us find the area under a curve, and in many cases, we use known area formulas to simplify our calculations.
In this exercise, we are looking at the integral of a linear function, which is the area under a straight line. This type of area can sometimes form standard geometric shapes, like trapezoids. The area formula for these shapes becomes very handy.
In this exercise, we are looking at the integral of a linear function, which is the area under a straight line. This type of area can sometimes form standard geometric shapes, like trapezoids. The area formula for these shapes becomes very handy.
- The general area formula for a trapezoid is: \( A = \frac{1}{2} (b_1 + b_2) \times h \).
- \( b_1 \) and \( b_2 \) stand for the lengths of the two parallel sides.
- \( h \) is the height, measured as the distance between these sides.
Trapezoid
The shape of a trapezoid is quite special in geometry, defined by having two parallel sides of different lengths. In the context of integrals, when a straight line function is involved, the area under the curve might form a trapezoid.
The straight line here, represented by the function \( 3t \), when plotted, creates a trapezoidal region between the line and the x-axis within the interval \((a, b)\).
The straight line here, represented by the function \( 3t \), when plotted, creates a trapezoidal region between the line and the x-axis within the interval \((a, b)\).
- The reason we see a trapezoid is because our line \( 3t \) doesn't change slope; it remains linear and steady.
- This stable rise over run creates identical width (or height in this context) between different y-values, essentially outlining two parallel sides.
Geometric Interpretation
Geometric interpretation gives a visual context to mathematical concepts. When we say to geometrically understand \( \int_{a}^{b} 3t \, dt \), we imply determining the area under the curve 3t using geometry.
Visualizing the integral as a shape under the line\( 3t \) from \( a \) to \( b \), greatly simplifies the problem.
If the straight line function is positive over the interval, it resembles a traditional trapezoid when visualized on a graph.
Visualizing the integral as a shape under the line\( 3t \) from \( a \) to \( b \), greatly simplifies the problem.
If the straight line function is positive over the interval, it resembles a traditional trapezoid when visualized on a graph.
- The line provides the inclined side, and the x-axis provides one of the parallel sides.
- In practice, this involves sketching the line and shading the area to find the geometric region in focus.
Straight Line Function
The function \( 3t \) is a typical example of a straight line function from elementary algebra. This function is linear, which means it graphs out as a straight line on a two-dimensional coordinate system.
When evaluating the integral of this line between two points, you are effectively calculating the area beneath it within those bounds.
When evaluating the integral of this line between two points, you are effectively calculating the area beneath it within those bounds.
- Linear functions have a representative equation \( y = mx + b \), where \( m \) is the slope and \( b \) the y-intercept.
- For \( 3t \), the slope \( m \) is 3, indicating that for every unit increase in \( t \), \( y \) rises by 3.
- There's no y-intercept since the function perfectly scales as the variable \( t \) and does not shift on the y-axis.
