The method of finding the area between curves is dependent on the technique of integration. It's essentially the act of figuring out the space that's captured within two or more curves plotted on a graph. These curves could intersect, resulting in various distinct regions. You can calculate this area by integrating the absolute difference between the functions that define these curves across a specified interval.
| Topic | Problem | Solution |
|---|---|---|
| None | Find the area of the region enclosed by the curve… | The area enclosed by two curves \(f(x)\) and \(g(x)\) from \(a\) to \(b\) is given by the integral … |
| None | Find the area of the region enclosed by the curve… | Set the two functions equal to each other to find the points of intersection: \(x^{2}-4 = 5\). |
| None | Find the area of the shaded region enclosed by th… | First, we need to find the points of intersection between the functions. This can be done by settin… |
| None | Find the total area of the shaded region bounded … | First, we need to find the intersection points of the two curves. We set \(8y^2 - 8y^3 = 6y^2 - 6y\… |
| None | a) Approximate the area under graph (a) of $f(x)=… | First, we need to understand that the area under the curve of a function can be approximated by the… |
| None | 2 pts Which choice below represents the area of t… | First, we need to find the intersection points of the two curves. Set \(r_{1} = r_{2}\), we get \(3… |
| None | Use the Fundamental Theorem of Calculus to find t… | First, we need to find the integral of the function \(y=-x^{2}+7 x\) from 2 to 6. This is done usin… |
| None | Find the area of the region bounded by the graphs… | We are given the equations \(y=x^{2}+5\), \(y=x^{2}\), \(x=0\), and \(x=3\). We are asked to find t… |
| None | Use finite approximations to estimate the area un… | We are given the function \(f(x)=4-x^{2}\) and we are asked to estimate the area under the curve of… |
| None | Find the area of the region bounded by the graphs… | We are given two equations, \(y=13x\) and \(y=x^{2}\). We are asked to find the area of the region … |
| None | Find the area of the region bounded by the graphs… | First, we need to find the points of intersection of the two curves. This is done by setting the tw… |
| None | Find the area under the given curve over the indi… | We are given the function \(y = x^{2} + x + 2\) and we need to find the area under the curve from \… |
| None | Find the area of the shaded region. \[ f(x)=3 x+2… | The area of the shaded region between two functions, f(x) and g(x), from a to b is given by the int… |
| None | Find the area under the given curve over the indi… | The problem is to find the area under the curve defined by the function \(y=e^{2x}\) over the inter… |
| None | Find the area of the region bounded by the graphs… | Set the two equations equal to each other to find the points of intersection: \(2x^{2} - 5x + 6 = x… |
| None | Approximate the area under the graph of $F(x)=0.9… | Determine the width of each subinterval. The total width of the interval is \(-2 - (-7) = 5\), so e… |
| None | Approximate the area under the graph of $f(x)=0.0… | We are given the function \(f(x)=0.05 x^{4}-1.44 x^{2}+66\) and asked to approximate the area under… |
| None | Evaluate the integral below by interpreting it in… | The integral of a function over an interval can be interpreted as the area under the curve of the f… |
| None | Estimate the area under the graph of $f(x)=\frac{… | Define the interval from a = 1 to b = 4 and the number of rectangles n = 12. |
| None | Question 1, 5.1.2 Part 3 of 4 Estimate the area u… | We are asked to estimate the area under the curve of the function \(f(x)=5x^3\) between \(x=0\) and… |
| None | Use finite approximation to estimate the area und… | Divide the interval [0,2] into two equal subintervals [0,1] and [1,2] for the lower sum with two re… |
| None | Find the area under the graph of $f$ over the int… | The function is defined piecewise, so we need to split the interval into two parts: [1,2] and (2,3]. |
| None | Approximate the area under the curve graphed belo… | Define the function that represents the curve. For this demonstration, let's assume the function is… |
| None | Approximate the area under the curve $y=x^{3}$ fr… | We are asked to approximate the area under the curve \(y=x^{3}\) from \(x=1\) to \(x=3\) using a Ri… |
| None | Find the area of the region under the graph of th… | The area under the curve of a function from a to b is given by the definite integral of the functio… |
| None | Find the area of the shaded region. \[ f(x)=10 x+… | The area of the shaded region between two functions, f(x) and g(x), from a to b is given by the int… |
| None | Find the area under the graph of $f$ over the int… | The function is a piecewise function, so we need to split the integral into two parts: one for the … |
| None | Find the area of the region bounded by the graphs… | First, we need to find the points of intersection of the two curves. This can be done by setting th… |
| None | Find the area of the region enclosed by $f(x)$ an… | The area enclosed by a function \(f(x)\) and the \(x\)-axis over an interval \([a, b]\) can be foun… |
| None | Calculate the area, in square units, of the regio… | First, we need to find the points of intersection of the line and the parabola. We set \(g(x) = f(x… |
| None | Find the area between the curve and the $x$-axis … | The problem is to find the area between the curve and the x-axis over the interval [1,6] for the fu… |
| None | Find the area under the graph of $f$ over the int… | The function is defined differently for different parts of the interval, so we will need to split t… |
| None | Find the area under the given curve over the indi… | The area under a curve from a to b is given by the definite integral from a to b of the function. I… |
| None | The graph above satisfy the equation $x^{4}+y^{3}… | First, we rewrite the equation $x^{4}+y^{3}=x^{2} y$ as $x^{4}-x^{2} y+y^{3}=0$. |