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Questions and Answers
What is the vertex of the parabola that models the river's function?
What is the approximate area of farmland calculated using the Trapezoidal Rule?
What is the correct interpretation of the sum of the trapeziums used to estimate area under the curve?
What is the exact area of productive farmland calculated using calculus?
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What is the calculated volume of the stadium?
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How many air-conditioners are required for the stadium, given their servicing capacity?
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What is the x-coordinate of point C on the parabola representing the stadium's roof?
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What is the formula to calculate the cross-sectional area ABCDE of the stadium?
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What is the result of the integral $ rac{1}{2} imes (9 + 26)$ from Question 4?
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For the function f(x) = $9 - x^2$, what are the x-intercepts?
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What is the value of C when determining f(x) given f'(x) = 6x - 3 and f(1) = 2?
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What is the area under the curve $y = x^2 + 2$ from x = 0 to x = 2?
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What is the result of the integral $igg( rac{x^3}{3} + 2xigg) |_0^2$?
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What is the integral expression that represents the area bound by the x-axis, x = -1, and x = 2 for the function f(x) = $1 - x^3$?
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Which of the following represents the anti-derivative of $2(x+2)^2$?
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What is the correct anti-derivative of $f'(x) = 6x - 3$?
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What is the result of $A = [ rac{x^4}{4} - x]_0^1 - [ rac{x^4}{4} - x]_1^2$?
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What is the correct antiderivative of the function $3x^2 - 3x + 1$?
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For the equation $y = 2x + 4$ when $x = 0$, what is the corresponding value of $y$?
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Which expression correctly represents $ rac{1}{3} imes ext{Area} - 1$ if $Area$ is defined as $ orall x ext{ in the range}$?
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What does the expression $ rac{5}{3}$ represent regarding the shaded area and curve on the graph?
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In the equation $y = k(x - 3)^2$, finding $k$ if the area under the curve is 36 yields which value?
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The expression for the total area of the shaded region between $x = 1$ and $x = 3$ with respect to $f(x)$ is which of the following?
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Study Notes
Anti-Differentiation
- $\int 6x^2 + 4x dx$ = $x^3 + 2x^2 + C$
- $\int 2(x+2)^2 dx$ = $\frac{2}{3}x^3 + 2x^2 + 8x + C$
- $\int \frac{x^3 + 3}{x^2} dx$ = $\frac{x^2}{2} + \frac{3}{x} + C$
- Find f(x) given f'(x) = 6x-3 and f(1) = 2: f(x) = $3x^2 - 3x + 2$
- Area under $y = x^2 + 2$ from x=0 to x=2: $\frac{20}{3}$ sq. units
- Trapezoidal estimate for area under $y = 9-x^2$ from x=0 to x = 3 is $\frac{35}{2}$ sq. units
- Area under the curve of $y = 1-x^3$ between x=-1 and x=2: $\frac{19}{4}$ sq. units
Multiple Choice
- Integral of $3x^2-3x+1$ is $x^3-3x^2+x+C$
- If $y = 2x + 4$ and $y = 1$ when $x = 0$, then $y= x^2 + 4x + 1$
- If $y = f(x)$ where f'(x) = $6x + k$ and has a stationary point at (1, 2), then $f(x) = 3x^2 - 6x + 5$
- $\int_0^a (2x + 4) dx = 12$ then a = 2
- $\int_{-1}^2 (3f(x) -1) dx$ is $3 \int_{-1}^2 f(x) dx - 1$
- Area under the curve between the line and curve is $\frac{9}{2}$ square units
- $y = k(x-3)^2$ has an area of 36 square units, then k = 4
- Total shaded area of a curve is $\int_1^3 f(x)dx - \int_3^4 f(x) dx$
Extended Response
Question 1
- Agricultural land is 10km x 10km, river is modelled by $r(x) = \frac{1}{10}(x-3)^2+8$
- Trapezoidal estimate for area under $r(x)$ is 67 km², underestimating the area
- Exact area of farmland is $\frac{203}{3}$ km²
- Farm is likely to be economically viable as ratio of productive farmland is greater than 2:1.
Question 2
- Parabolic shaped roof of a stadium is modelled by $y = 20 - \frac{3}{125}x^2$:
- Point E coordinates: (0, 20)
- Point C coordinates: (25, 5)
- Cross-sectional area of stadium is approximately 750 square metres
- Volume of stadium is 56,250 m³
- Number of air conditioners needed is 5.
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Description
Test your understanding of anti-differentiation and area under curves with this quiz. It covers various types of integrals, finding functions from derivatives, and calculating areas. Perfect for students revising calculus concepts.
