answer:Li Yi woke up to the sound of her mother's coughing. She looked at the clock and realized it was already past midnight. She sighed deeply as she got up from her bed and made her way to her mother's room. She had just gone to bed a few hours ago, but it seemed like her sleep was going to be interrupted again. As she entered the room, the coughing subsided, and her mother was gasping for air. Li Yi knew exactly what to do; she quickly grabbed the inhaler from the bedside table and helped her mother breathe. Her mother's eyes met hers, and Li Yi could see the gratitude in them. Thank you, my daughter, her mother whispered. I'm sorry to have disturbed you again. Li Yi smiled and patted her mother's hand. Don't apologize, Ma. I'm here for you. Taking care of her elderly mother was just one of the many responsibilities that Li Yi had to bear. As a single mother, she had to provide for her young daughter, Xiao Mei, as well. Her ex-husband had abandoned them both, leaving them to fend for themselves. Li Yi had been working two jobs to make ends meet, and she was constantly exhausted. But she knew she had to keep going. Her mother's illness required constant attention and medication, and Xiao Mei needed a stable home and a good education. Li Yi didn't have time to think about her own needs and desires; she was focused on taking care of her family. As she sat next to her mother, watching her breathe easier, Li Yi couldn't help but wonder if she was doing enough. She felt guilty for being too tired to spend more quality time with Xiao Mei, for not having the means to provide better healthcare for her mother. But she quickly pushed those thoughts away. She had to be strong for her family, no matter what. Li Yi leaned back in the chair, closing her eyes for a brief moment of respite. She knew tomorrow would be another long day, filled with challenges and sacrifices. But she was ready to face them head-on, for the sake of the people she loved.

answer:When finding the area between a curve and the x-axis, we typically use a definite integral. The definite integral of a function f(x) with respect to x, denoted as ∫[a, b] f(x) dx, represents the area between the curve y = f(x) and the x-axis within the interval [a, b]. More specifically, if f(x) is a continuous function on the interval [a, b], then the definite integral ∫[a, b] f(x) dx is equal to the area of the region bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b. If the curve lies above the x-axis, the definite integral gives the area between the curve and the x-axis. If the curve lies below the x-axis, the definite integral gives the negative of the area. This concept is often referred to as the Fundamental Theorem of Calculus, which establishes a deep connection between differentiation and integration. It's a fundamental tool for evaluating definite integrals and has far-reaching implications in many areas of mathematics and physics.

answer:A double integral is an extension of the definite integral to two dimensions. It's used to find the volume under a surface defined by a function of two variables, z = f(x, y), over a region in the xy-plane. The double integral represents the volume of the solid bounded by the surface, the xy-plane, and the boundary of the region. The double integral of a function f(x, y) over a region R is denoted as ∫∫R f(x, y) dA or ∫∫R f(x, y) dx dy. To evaluate a double integral, we need to iterate the integration process twice, first with respect to one variable and then with respect to the other variable. One way to think about it is to imagine the region R being divided into infinitesimal rectangles, each with area dA = dx dy. The double integral then sums up the volumes of the infinitesimal boxes formed by these rectangles, where the height of each box is given by the function value f(x, y). There are two main approaches to evaluating double integrals: the iterated integral approach and the change of variables approach. The iterated integral approach involves setting up two separate integrals, one with respect to x and one with respect to y, and then evaluating them in succession. The order of integration can often be interchanged, but this may affect the complexity of the calculation. The change of variables approach involves transforming the region R into a simpler region, often using polar coordinates or other coordinate systems, and then evaluating the integral in the new coordinate system. This approach can greatly simplify certain types of double integrals. Double integrals have many applications in physics, engineering, and other fields, such as calculating the center of mass of an object, the moment of inertia, or the volume of a complex shape.

answer:When dealing with a circular domain, switching to polar coordinates can greatly simplify the evaluation of a double integral. In polar coordinates, a point in the xy-plane is represented by (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. When converting a double integral from Cartesian coordinates (x, y) to polar coordinates (r, θ), we need to make two changes: 1. The differential area element dA = dx dy becomes dA = r dr dθ. This is because the area of a small sector of a circle is proportional to the radius r. 2. The region of integration changes from a rectangle in the xy-plane to a rectangle in the rθ-plane. The double integral in polar coordinates becomes ∫∫R f(r cos θ, r sin θ) r dr dθ. Here, we've replaced x with r cos θ and y with r sin θ, using the relationships between Cartesian and polar coordinates. The limits of integration also change. For a circular domain, the limits for r typically range from 0 to a (the radius of the circle), and the limits for θ range from 0 to 2π (or from 0 to π, depending on the specific region). Using polar coordinates can simplify the integral in several ways: * The circular boundary becomes a simple rectangular boundary in the rθ-plane. * The Jacobian factor r in the differential area element can simplify the integrand. * Trigonometric functions can be simplified using identities involving r and θ. For example, if we have a function f(x, y) = x^2 + y^2 and we want to integrate it over a circle of radius a, the double integral in Cartesian coordinates would be ∫∫R (x^2 + y^2) dx dy. In polar coordinates, this becomes ∫∫R r^2 r dr dθ, which is much simpler to evaluate. Overall, using polar coordinates can greatly simplify double integrals with circular domains, making it easier to evaluate these types of integrals.