Finally, we evaluate the definite integral of the line equation from x=0 to x=7. This calculates the area under the line segment PQ, which is the area of the triangle POQ. This confirms the standard formula for the area of a right-angled triangle: 21×base×height=21×7×6=21.
Area of △POQ=76∫07(7−x)dx
To integrate the function 76(7−x) with respect to x, we can first take the constant 76 out of the integral sign. This is a property of integrals: ∫c⋅f(x)dx=c⋅∫f(x)dx, where 'c' is a constant. This simplifies the integration process, allowing us to focus on integrating the polynomial (7−x).
76[7x−2x2]07
Now we integrate the expression (7−x) term by term. The integral of a constant, say 'a', with respect to x is ax. So, the integral of 7 is 7x. The integral of x (which is x1) with respect to x is 1+1x1+1=2x2. (Recall the power rule for integration: ∫xndx=n+1xn+1+C). Since we are dealing with a definite integral, we don't add the constant of integration, C. After integrating, we denote the limits of integration from 0 to 7 using the square bracket notation.
76[(7(7)−272)−(7(0)−202)]=76[49−249−0]=76[298−49]=76[249]=3⋅7=21 units2
This step involves evaluating the definite integral using the Fundamental Theorem of Calculus. First, we substitute the upper limit (x=7) into the integrated expression: 7(7)−272=49−249. Next, we substitute the lower limit (x=0) and get 7(0)−202=0. We then subtract the result of the lower limit substitution from the result of the upper limit substitution: (49−249)−0. To simplify 49−249, we find a common denominator, which is 2. So, 49=298. Thus, 298−249=249. Finally, multiply this result by the constant 76: 76×249. We can simplify this by cancelling out common factors: 7 divides 49 to give 7, and 2 divides 6 to give 3. So, 3×7=21. This gives us the area of the triangle POQ as 21 square units. (definite integral, Fundamental Theorem of Calculus, area, right-angled triangle, power rule for integration)
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Step-by-Step Solution
Step 1: Introduce the integral for the area
The problem asks us to evaluate the given definite integral to find the area of the triangle POQ. The expression 76(7−x) represents the equation of the line segment whose area we are calculating from x=0 to x=7.
Step 2: Integrate the function
To integrate the function, we apply the power rule of integration: ∫xndx=n+1xn+1. For the constant term, ∫7dx=7x. For the term −x, we integrate it as −2x2. The constant 76 is kept outside the integral and applied after evaluating the definite integral.
Step 3: Evaluate the definite integral
We apply the Fundamental Theorem of Calculus by substituting the upper limit (x=7) and the lower limit (x=0) into the integrated expression. First, substitute x=7 to get 7(7)−272=49−249. Then, substitute x=0 to get 7(0)−202=0. Subtract the lower limit result from the upper limit result: (49−249)−0. Simplify 49−249=298−249=249. Finally, multiply by the constant 76: 76×249=73×49=3×7=21. This yields the area of the triangle as 21 square units.