Area Under a Curve: Basics, Units & Intro to Riemann Sums

The area under a rate of change function over an interval gives the accumulation of change over that interval. Here, the rate is in gallons per minute and time is in minutes. The area is found by multiplying gallons per minute by minutes, which yields gallons. This represents the total change in volume from t=0 to t=10.

The region under f(x)=4 from x=1 to x=5 is a rectangle. The width of the rectangle is 5 - 1 = 4. The height is 4. The area of a rectangle is width multiplied by height. So, the area is 4 * 4 = 16.

If the rate of change is negative over an interval, then the accumulation of change, which is represented by the area under the curve, is negative. This means that the quantity is decreasing over that interval.

The unit of the area is the unit of the rate of change multiplied by the unit of the independent variable. Here, rate is in meters per second and time is in seconds. Multiplying meters per second by seconds gives meters. This represents distance.

The trapezoidal rule with one trapezoid gives the area of a trapezoid. The bases are f(0)=2 and f(3)=8. The height is 3 - 0 = 3. The area of a trapezoid is (½) times the sum of the bases times the height. So, area = (½)*(2+8)*3 = (½)*10*3 = 15.

The interval [0,3] is divided into 3 subintervals of equal width. The width of each subinterval is (3-0)/3 = 1. The left endpoints are 0, 1, and 2. The function values at these points are f(0)=0, f(1)=1, and f(2)=4. The left Riemann sum is the sum of the areas of rectangles: width times the sum of the function values. So, sum = 1*(0+1+4) = 5.

For a decreasing function, the function value at the left endpoint of each subinterval is the maximum value on that subinterval. The left Riemann sum uses these left endpoint values, so it gives an overestimate of the area under the curve.

The given sum is a Riemann sum. The term (3i/n) suggests the sample points are xᵢ = 3i/n. As i goes from 1 to n, xᵢ ranges from 3/n to 3. In the limit, this covers the interval [0,3]. The function is (xᵢ)², so it is x². The width of each subinterval is 3/n. Therefore, the limit represents the definite integral of x² from 0 to 3.

With 4 subintervals, the width is (2-0)/4 = 0.5. The x values are 0, 0.5, 1, 1.5, 2. The function values are f(0)=0, f(0.5)=0.25, f(1)=1, f(1.5)=2.25, f(2)=4. The trapezoidal rule formula is (width/2) times [f(first) + 2*(sum of interior values) + f(last)]. So, area = (0.5/2)*[0 + 2*(0.25+1+2.25) + 4] = 0.25*[0 + 2*(3.5) + 4] = 0.25*[0+7+4] = 0.25*11 = 2.75.

A Riemann sum is defined as the sum of products of function values and the widths of subintervals. Option B exactly matches this definition: f(xᵢ) is the function value at a point in the subinterval, and (xᵢ - x_{i-1}) is the width of that subinterval.

The formal definition of a definite integral is the limit of Riemann sums as the number of subintervals approaches infinity and the width of each subinterval approaches zero.

The area is given by the integral from 0 to 4 of √(x) dx. The antiderivative of √(x) = x^(½) is (⅔)x^(3/2). Evaluating from 0 to 4: (⅔)*4^(3/2) - (⅔)*0^(3/2) = (⅔)*8 - 0 = 16/3.

The function sin(x) is positive on [0, pi]. The area is given by the definite integral from 0 toπ of sin(x) dx. The antiderivative of sin(x) is -cos(x). Evaluating from 0 to pi: -cos(pi) - (-cos(0)) = -(-1) + 1 = 1+1=2.

The curves intersect at x=0 and x=1. On [0,1], the line y=x is above y=x². The area is given by the ∫₀¹ (x - x²) dx. The antiderivative is (½)x² - (⅓)x³. Evaluating from 0 to 1: (1/2 -⅓) - (0) = (3/6 - 2/6) = 1/6.

The function x³ is odd, so the integral from -1 to 1 is 0. However, the area is the total area enclosed, which requires taking absolute values. We split the integral at 0. From -1 to 0, x³ is negative, so the area contribution is the integral from -1 to 0 of -x³ dx. From 0 to 1, x³ is positive, so the area is the ∫₀¹ x³ dx. The antiderivative of x³ is (1/4)x⁴. For the first part: -[(1/4)(0)⁴ - (1/4)(-1)⁴] = -[0 - 1/4] = 1/4. For the second part: (1/4)(1)⁴ - (1/4)(0)⁴ = 1/4. Total area = 1/4 + 1/4 = 1/2.