Squaring
1.Explain the term “quadrature”
Solution:
Quadrature is the calculation of the area of a plane figure that is bounded by the x-axis, two lines parallel to the y-axis at the points x1 = a, x2 = b, and a line (straight or curved) with the equation y = f(x).
2.Using a definite integral, derive the formula for calculating the area of
- a) a square
- b) a rectangle
Solution:
a) Square
The square is bounded by the x-axis, the line y = a, and lines parallel to the y-axis at the points a = 0, b = a.
b) Rectangle
The rectangle is bounded by the x-axis, the line y = b, and lines parallel to the y-axis at the points a = 0, b = a.
3.Using a definite integral, derive the formula for calculating the area of
- a) an isosceles triangle
- b) an isosceles trapezoid
Solution:
a) Isosceles triangle
The isosceles triangle has a base a and height v. Let S1 be half of the area of this triangle, which is bounded by the x-axis, lines parallel to the y-axis at the points a = 0, b = v, and the line
b) Isosceles trapezoid
The isosceles trapezoid has an upper base a, a lower base b, and height v. Let S1 be half of the area of this trapezoid, which is bounded by the x-axis, lines parallel to the y-axis at the points a = 0, b = v, and the line
4.Calculate the area of the figure bounded by the parabola y = x2 + 1 and the line y = 5.
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5.Using a definite integral, calculate the area of the plane figure bounded by the parabolas
y = 6 – 4x + x2 and y = –3 + 8x – 2x2.
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6. Derive the formula for calculating the area of a circle with radius r.
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